Skip to main content
뒤로

Comprehensive College Algebra & Precalculus Study Guide

스터디 가이드 - 스마트 노트

자료에 맞춘 맞춤형 노트, 핵심 정의, 예시, 맥락을 확장해 제공합니다.

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Functions and Their Graphs"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Definitions and Properties"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"A "},{"type":"text","marks":[{"type":"bold"}],"text":"function"},{"type":"text","text":" is a relation where each input (x) has exactly one output (y), denoted as f(x). Understanding the domain and range is essential for analyzing functions."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" All allowable x-values (inputs). Exclude values that make denominators zero or produce negative radicands for even roots."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Range:"},{"type":"text","text":" All possible y-values (outputs)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Interval Notation:"},{"type":"text","text":" Use ( ) for open intervals (excluding endpoints), [ ] for closed intervals (including endpoints), and ∪ for unions."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Common Function Types"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Type"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"General Form"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Key Feature"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = mx + b"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Constant slope"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = ax^2 + bx + c"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Parabola; vertex at "},{"type":"inlineMath","attrs":{"latex":"\\left(\\frac{-b}{2a}, f\\left(\\frac{-b}{2a}\\right)\\right)"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Polynomial"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = a_nx^n + ... + a_0"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Smooth, continuous curve"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = p(x)/q(x)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Has asymptotes; exclude q(x) = 0"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Radical"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = \\sqrt[n]{x}"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Even root: domain x ≥ 0"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Absolute Value"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) = |x|"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"V-shaped graph; vertex at origin"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Piecewise"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Different formulas on intervals"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Evaluate by matching interval"}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Even vs. Odd Functions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Even:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(-x) = f(x)"}},{"type":"text","text":" (symmetric about the y-axis)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Odd:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(-x) = -f(x)"}},{"type":"text","text":" (symmetric about the origin)"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Transformations of f(x)"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Transformation"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Equation"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Effect"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical shift up"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) + k"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Up k units"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical shift down"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x) - k"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Down k units"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal shift right"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x - h)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Right h units"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal shift left"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(x + h)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Left h units"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reflect over x-axis"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"-f(x)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Flips vertically"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reflect over y-axis"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(-x)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Flips horizontally"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical stretch"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a·f(x), |a| > 1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Taller by factor a"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Vertical compression"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"a·f(x), 0 < |a| < 1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Shorter by factor a"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal stretch"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(bx), 0 < b < 1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Wider by factor 1/b"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal compression"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"f(bx), b > 1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Narrower by factor 1/b"}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Composition & Inverse Functions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Composition:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"(f \\circ g)(x) = f(g(x))"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Inverse Function:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":" exists if f is one-to-one (passes the Horizontal Line Test). To find $f^{-1}(x)$: replace f(x) with y, swap x and y, solve for y, and write as $f^{-1}(x)$."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Key Property:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(f^{-1}(x)) = x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(f(x)) = x"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Polynomial and Rational Functions"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Polynomial Basics"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Degree (n):"},{"type":"text","text":" Highest power of x; determines end behavior and maximum number of turns (n−1)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Leading Coefficient (a_n):"},{"type":"text","text":" Sign determines end behavior direction."}]}]}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degree"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Leading Coefficient"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Left End"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Right End"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Even"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Positive (+)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Up"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Up"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Even"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative (−)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Down"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Down"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Odd"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Positive (+)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Down"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Up"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Odd"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Negative (−)"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Up"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Down"}]}]}]}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Zeros & Multiplicity:"},{"type":"text","text":" Odd multiplicity: graph crosses x-axis; even multiplicity: graph touches and turns."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Quadratic Formula:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Discriminant:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"b^2 - 4ac"}},{"type":"text","text":" (determines number and type of roots)"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Key Theorems"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Theorem"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Statement"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Remainder Theorem"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Dividing f(x) by (x − c) gives remainder f(c)"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Factor Theorem"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(x − c) is a factor of f(x) iff f(c) = 0"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Rational Root Theorem"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Possible rational roots: ±(factors of constant term)/(factors of leading coefficient)"}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Synthetic Division (Steps)"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write c in the box; list all coefficients of f(x) (use 0 for missing terms)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Bring down the first coefficient."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply by c; write result under next coefficient; add."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Repeat; last number is remainder, others are quotient coefficients."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Rational Functions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Vertical Asymptotes (VA):"},{"type":"text","text":" Set denominator q(x) = 0 (where factors do not cancel)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Holes:"},{"type":"text","text":" Occur where a factor cancels from numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Horizontal Asymptotes (HA):"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"deg(p) < deg(q): HA at y = 0"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"deg(p) = deg(q): HA at y = (leading coeff. of p)/(leading coeff. of q)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"deg(p) > deg(q): No HA (may have oblique asymptote)"}]}]}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Oblique Asymptote:"},{"type":"text","text":" Perform polynomial division; quotient (ignoring remainder) is the asymptote."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Exponential and Logarithmic Functions"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Exponential Functions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Standard Form:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(x) = a \\cdot b^x"}},{"type":"text","text":" (a ≠ 0, b > 0, b ≠ 1)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Growth:"},{"type":"text","text":" b > 1; "},{"type":"text","marks":[{"type":"bold"}],"text":"Decay:"},{"type":"text","text":" 0 < b < 1"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Natural Exponential:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"f(x) = e^x"}},{"type":"text","text":" (e ≈ 2.71828)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" (−∞, ∞); "},{"type":"text","marks":[{"type":"bold"}],"text":"Range:"},{"type":"text","text":" (0, ∞); "},{"type":"text","marks":[{"type":"bold"}],"text":"HA:"},{"type":"text","text":" y = 0"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Logarithmic Functions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Definition:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"y = \\log_a(x) \\iff a^y = x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Natural Log:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\ln(x) = \\log_e(x)"}},{"type":"text","text":"; "},{"type":"text","marks":[{"type":"bold"}],"text":"Common Log:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\log(x) = \\log_{10}(x)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Domain:"},{"type":"text","text":" (0, ∞); "},{"type":"text","marks":[{"type":"bold"}],"text":"Range:"},{"type":"text","text":" (−∞, ∞); "},{"type":"text","marks":[{"type":"bold"}],"text":"VA:"},{"type":"text","text":" x = 0"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Properties of Logarithms"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Property"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Formula"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Product Rule"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(MN) = \\log_a(M) + \\log_a(N)"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient Rule"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(M/N) = \\log_a(M) - \\log_a(N)"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Power Rule"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(M^p) = p \\cdot \\log_a(M)"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Change of Base"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(x) = \\frac{\\log(x)}{\\log(a)} = \\frac{\\ln(x)}{\\ln(a)}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse Properties"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(a^x) = x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"a^{\\log_a(x)} = x"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Log of 1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(1) = 0"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Log of base"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\log_a(a) = 1"}}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Solving Equations"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Exponential:"},{"type":"text","text":" Isolate the exponential, then take log of both sides. Example: "},{"type":"inlineMath","attrs":{"latex":"3^x = 20 \\implies x = \\frac{\\ln 20}{\\ln 3}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Logarithmic:"},{"type":"text","text":" Condense to single log, convert to exponential form. Example: "},{"type":"inlineMath","attrs":{"latex":"\\log_2(x) = 5 \\implies x = 2^5 = 32"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Check for extraneous solutions:"},{"type":"text","text":" Argument of any log must be > 0."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Financial Applications"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Compound Interest:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"A = P(1 + \\frac{r}{n})^{nt}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Continuous Compounding:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"A = Pe^{rt}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"P = principal, r = annual rate (decimal), n = compoundings/year, t = years"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Trigonometry: Angles and the Unit Circle"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Angle Conversions"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi"}},{"type":"text","text":" radians = 180°"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degrees to radians: Degrees × "},{"type":"inlineMath","attrs":{"latex":"(\\pi/180)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Radians to degrees: Radians × "},{"type":"inlineMath","attrs":{"latex":"(180/\\pi)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Arc length:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"s = r\\theta"}},{"type":"text","text":" (θ in radians)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Sector area:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"A = \\frac{1}{2}r^2\\theta"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Unit Circle Reference Table"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Degrees"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Radians"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cos θ"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sin θ"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"tan θ"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0°"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"30°"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi/6"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"1/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}/3"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"45°"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi/4"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{2}/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{2}/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"60°"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi/3"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"1/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sqrt{3}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"90°"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi/2"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"0"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Undefined"}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Six Trig Functions (Unit Circle Definitions)"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Unit Circle"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reciprocal Form"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sin θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y-coordinate"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"—"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cos θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"x-coordinate"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"—"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"tan θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y/x"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{\\sin \\theta}{\\cos \\theta}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"csc θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1/y"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{1}{\\sin \\theta}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"sec θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1/x"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{1}{\\cos \\theta}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"cot θ"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"x/y"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\frac{\\cos \\theta}{\\sin \\theta}"}}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"ASTC Sign Rules by Quadrant"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"QI (0° to 90°): All functions positive"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"QII (90° to 180°): Sine (and csc) positive only"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"QIII (180° to 270°): Tangent (and cot) positive only"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"QIV (270° to 360°): Cosine (and sec) positive only"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Reference and Coterminal Angles"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Reference angle:"},{"type":"text","text":" Acute angle between terminal side and x-axis; always positive and between 0° and 90°."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Coterminal angles:"},{"type":"text","text":" Add or subtract 360° (or 2π) any number of times."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Trigonometric Functions and Graphs"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"General Sinusoidal Form"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y = A \\sin(Bx - C) + D"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"y = A \\cos(Bx - C) + D"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Amplitude:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"|A|"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Period:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"2\\pi/|B|"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Phase Shift:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"C/B"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Vertical Shift:"},{"type":"text","text":" D"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Midline:"},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"y = D"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Key Graph Features"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Amplitude"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Period"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Asymptotes?"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y = sin x"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"2\\pi"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(−\\infty, \\infty)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[−1, 1]"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"No"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y = cos x"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"1"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"2\\pi"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(−\\infty, \\infty)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[−1, 1]"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"No"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"y = tan x"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"None"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\pi"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x \\neq \\frac{\\pi}{2} + n\\pi"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(−\\infty, \\infty)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Yes"}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Inverse Trig Functions — Domain & Range"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Function"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Notation"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Domain"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse Sine"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"arcsin(x) or "},{"type":"inlineMath","attrs":{"latex":"\\sin^{-1}(x)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[−1, 1]"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[−\\frac{\\pi}{2}, \\frac{\\pi}{2}]"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse Cosine"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"arccos(x) or "},{"type":"inlineMath","attrs":{"latex":"\\cos^{-1}(x)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[−1, 1]"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse Tangent"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"arctan(x) or "},{"type":"inlineMath","attrs":{"latex":"\\tan^{-1}(x)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(−\\infty, \\infty)"}}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"(−\\frac{\\pi}{2}, \\frac{\\pi}{2})"}}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Trigonometric Identities"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"Fundamental Identities"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Category"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identities"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Pythagorean"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^2\\theta + \\cos^2\\theta = 1"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"1 + \\tan^2\\theta = \\sec^2\\theta"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"1 + \\cot^2\\theta = \\csc^2\\theta"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reciprocal"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\csc \\theta = 1/\\sin \\theta"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = 1/\\cos \\theta"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = 1/\\tan \\theta"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quotient"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = \\frac{\\sin \\theta}{\\cos \\theta}"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = \\frac{\\cos \\theta}{\\sin \\theta}"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Even/Odd"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos(-\\theta) = \\cos \\theta"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\sin(-\\theta) = -\\sin \\theta"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\tan(-\\theta) = -\\tan \\theta"}}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Co-function"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin \\theta = \\cos(\\frac{\\pi}{2} - \\theta)"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\tan \\theta = \\cot(\\frac{\\pi}{2} - \\theta)"}},{"type":"text","text":" "},{"type":"inlineMath","attrs":{"latex":"\\sec \\theta = \\csc(\\frac{\\pi}{2} - \\theta)"}}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Sum & Difference Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(A \\pm B) = \\sin A \\cos B \\pm \\cos A \\sin B"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos(A \\pm B) = \\cos A \\cos B \\mp \\sin A \\sin B"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(A \\pm B) = \\frac{\\tan A \\pm \\tan B}{1 \\mp \\tan A \\tan B}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Double and Half Angle Formulas"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(2\\theta) = 2 \\sin \\theta \\cos \\theta"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos(2\\theta) = \\cos^2\\theta - \\sin^2\\theta = 1 - 2\\sin^2\\theta = 2\\cos^2\\theta - 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(2\\theta) = \\frac{2 \\tan \\theta}{1 - \\tan^2\\theta}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin(\\theta/2) = \\pm \\sqrt{\\frac{1 - \\cos \\theta}{2}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos(\\theta/2) = \\pm \\sqrt{\\frac{1 + \\cos \\theta}{2}}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan(\\theta/2) = \\pm \\sqrt{\\frac{1 - \\cos \\theta}{1 + \\cos \\theta}} = \\frac{\\sin \\theta}{1 + \\cos \\theta} = \\frac{1 - \\cos \\theta}{\\sin \\theta}"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":2},"content":[{"type":"text","text":"Solving Trigonometric Equations"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","text":"General Process"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate the trigonometric function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the reference angle using inverse trig functions."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine all solutions in the given interval using the unit circle and ASTC rules."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the general solution by adding the appropriate period (2π for sin/cos, π for tan)."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","text":"Equation Types and Strategies"}]},{"type":"table","content":[{"type":"tableRow","content":[{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Equation Type"}]}]},{"type":"tableHeader","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Strategy"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Linear (e.g., "},{"type":"inlineMath","attrs":{"latex":"2\\sin x - 1 = 0"}},{"type":"text","text":")"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Isolate sin x, find reference angle, apply ASTC, list solutions"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadratic (e.g., "},{"type":"inlineMath","attrs":{"latex":"2\\sin^2x - \\sin x - 1 = 0"}},{"type":"text","text":")"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Let "},{"type":"inlineMath","attrs":{"latex":"u = \\sin x"}},{"type":"text","text":", factor, solve each"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"With double angles (e.g., "},{"type":"inlineMath","attrs":{"latex":"\\sin(2x) = \\sqrt{3}/2"}},{"type":"text","text":")"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use double angle identity, solve for 2x, then divide by 2"}]}]}]},{"type":"tableRow","content":[{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Requires identity (e.g., "},{"type":"inlineMath","attrs":{"latex":"1 - \\cos^2x = \\sin x"}},{"type":"text","text":")"}]}]},{"type":"tableCell","attrs":{"colspan":1,"rowspan":1,"colwidth":null},"content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Replace "},{"type":"inlineMath","attrs":{"latex":"1 - \\cos^2x"}},{"type":"text","text":" with "},{"type":"inlineMath","attrs":{"latex":"\\sin^2x"}},{"type":"text","text":", factor, solve"}]}]}]}]},{"type":"heading"

Pearson Logo

스터디 프렙