Precalculus Trigonometry and Inverse Functions Study Guide

{"type":"doc","content":[{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. (a) Express 96 degrees in radians."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Degree-Radian Conversion"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert angles from degrees to radians, which is essential for working with trigonometric functions in calculus and precalculus."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key formula:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To convert degrees to radians:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\text{Radians} = \\text{Degrees} \\times \\frac{\\pi}{180}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the given angle in degrees: "},{"type":"inlineMath","attrs":{"latex":"96^\\circ"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the conversion using the formula: "},{"type":"inlineMath","attrs":{"latex":"96^\\circ \\times \\frac{\\pi}{180}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Simplify the fraction "},{"type":"inlineMath","attrs":{"latex":"\\frac{96}{180}"}},{"type":"text","text":" to lowest terms."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express your answer as a multiple of "},{"type":"inlineMath","attrs":{"latex":"\\pi"}},{"type":"text","text":" in radians."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q1. (b) Express "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{12}"}},{"type":"text","marks":[{"type":"bold"}],"text":" radians in degrees."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Radian-Degree Conversion"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to convert angles from radians to degrees, which is important for interpreting trigonometric values in different contexts."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key formula:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To convert radians to degrees:"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\text{Degrees} = \\text{Radians} \\times \\frac{180}{\\pi}"}}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the given angle in radians: "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{12}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Set up the conversion: "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{12} \\times \\frac{180}{\\pi}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Notice that "},{"type":"inlineMath","attrs":{"latex":"\\pi"}},{"type":"text","text":" cancels out in the numerator and denominator."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply "},{"type":"inlineMath","attrs":{"latex":"\\frac{5}{12}"}},{"type":"text","text":" by $180$ to get the degree measure."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q2. Evaluate the following trigonometric functions:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin\\left(\\frac{2\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{2\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan\\left(\\frac{2\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot\\left(\\frac{2\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin\\left(\\frac{5\\pi}{4}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{5\\pi}{4}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec\\left(\\frac{5\\pi}{4}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\csc\\left(\\frac{5\\pi}{4}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin\\left(\\frac{7\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{7\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot\\left(\\frac{7\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec\\left(\\frac{7\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin\\left(-\\frac{27\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{28\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot\\left(-\\frac{17\\pi}{3}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sec\\left(\\frac{29\\pi}{6}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^{-1}\\left(-\\frac{1}{2}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1}\\left(\\frac{1}{2}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan^{-1}(\\sqrt{3})"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan^{-1}\\left(-\\frac{1}{\\sqrt{3}}\\right)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^{-1}(-1)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1}(0)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1}\\left(-\\frac{\\sqrt{3}}{2}\\right)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Trigonometric and Inverse Trigonometric Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to evaluate trigonometric functions at special angles and to use the unit circle and reference angles. It also checks your understanding of inverse trigonometric functions and their ranges."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Unit Circle values for "},{"type":"inlineMath","attrs":{"latex":"\\sin"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cos"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\tan"}},{"type":"text","text":" at common angles (e.g., "},{"type":"inlineMath","attrs":{"latex":"30^\\circ"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"45^\\circ"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"60^\\circ"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\pi}{6}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\pi}{4}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\pi}{3}"}},{"type":"text","text":", etc.)"}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reciprocal identities: "},{"type":"inlineMath","attrs":{"latex":"\\csc x = \\frac{1}{\\sin x}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sec x = \\frac{1}{\\cos x}"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cot x = \\frac{1}{\\tan x}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse trigonometric function ranges:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^{-1} x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"[-\\frac{\\pi}{2}, \\frac{\\pi}{2}]"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos^{-1} x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"[0, \\pi]"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\tan^{-1} x"}},{"type":"text","text":": "},{"type":"inlineMath","attrs":{"latex":"(-\\frac{\\pi}{2}, \\frac{\\pi}{2})"}}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each angle, first reduce it to an equivalent angle between $0"},{"type":"inlineMath","attrs":{"latex":" and $2\\pi"}},{"type":"text","text":" if necessary (use coterminal angles)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the reference angle and the quadrant in which the terminal side lies."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall the unit circle values for "},{"type":"inlineMath","attrs":{"latex":"\\sin"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cos"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\tan"}},{"type":"text","text":" at the reference angle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the correct sign based on the quadrant."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For reciprocal functions ("},{"type":"inlineMath","attrs":{"latex":"\\csc"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sec"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cot"}},{"type":"text","text":"), use the reciprocal of the corresponding basic function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For inverse functions, recall the principal value ranges and find the angle whose trigonometric value matches the given number."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. (a) If a ray anchored at (0,0) and ending at (1,0) is rotated through "},{"type":"inlineMath","attrs":{"latex":"\\frac{27\\pi}{4}"}},{"type":"text","marks":[{"type":"bold"}],"text":" radians, in which quadrant does the particle end up?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Coterminal Angles and Quadrants"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to find the terminal position of an angle by reducing it to an equivalent angle between $0"},{"type":"inlineMath","attrs":{"latex":" and $2\\pi"}},{"type":"text","text":", and then determining the corresponding quadrant."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Coterminal angles: "},{"type":"inlineMath","attrs":{"latex":"\\theta_{\\text{coterminal}} = \\theta \\pm 2\\pi n"}},{"type":"text","text":" for integer "},{"type":"inlineMath","attrs":{"latex":"n"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Quadrant determination based on angle measure"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide "},{"type":"inlineMath","attrs":{"latex":"\\frac{27\\pi}{4}"}},{"type":"text","text":" by "},{"type":"inlineMath","attrs":{"latex":"2\\pi"}},{"type":"text","text":" to find how many full rotations are completed."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Subtract the integer multiple of "},{"type":"inlineMath","attrs":{"latex":"2\\pi"}},{"type":"text","text":" to find the coterminal angle between $0"},{"type":"inlineMath","attrs":{"latex":" and $2\\pi"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine in which quadrant this angle lies."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q3. (b) At which point (x, y) of the unit circle does the particle end up after this rotation?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Unit Circle Coordinates"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the coordinates on the unit circle corresponding to a given angle, using reference angles and quadrant signs."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Coordinates on the unit circle: "},{"type":"inlineMath","attrs":{"latex":"(\\cos \\theta, \\sin \\theta)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reference angle and quadrant sign rules"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the coterminal angle found in part (a)."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Determine the reference angle and its corresponding unit circle coordinates."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Apply the correct signs for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" based on the quadrant."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q4. If "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta > 0"}},{"type":"text","marks":[{"type":"bold"}],"text":" and "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta < 0"}},{"type":"text","marks":[{"type":"bold"}],"text":", in which quadrant does "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"type":"text","marks":[{"type":"bold"}],"text":" lie?"}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Signs of Trigonometric Functions in Quadrants"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of the signs of trigonometric functions in each quadrant and how to use inequalities to determine the quadrant of an angle."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cos \\theta > 0"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"\\theta"}},{"type":"text","text":" is in Quadrant I or IV."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\cot \\theta < 0"}},{"type":"text","text":" means "},{"type":"inlineMath","attrs":{"latex":"\\tan \\theta > 0"}},{"type":"text","text":" (since "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta = 1/\\tan \\theta"}},{"type":"text","text":"), so "},{"type":"inlineMath","attrs":{"latex":"\\tan \\theta"}},{"type":"text","text":" is negative in Quadrant II or IV."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the quadrants where "},{"type":"inlineMath","attrs":{"latex":"\\cos \\theta"}},{"type":"text","text":" is positive."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"List the quadrants where "},{"type":"inlineMath","attrs":{"latex":"\\cot \\theta"}},{"type":"text","text":" is negative."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the intersection of these two sets to determine the correct quadrant."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q5. Find the values of the 6 trigonometric functions evaluated at "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{6}"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Evaluating Trigonometric Functions at Special Angles"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find all six trigonometric function values for a given angle, using the unit circle and reference angles."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Unit circle values for "},{"type":"inlineMath","attrs":{"latex":"\\sin"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cos"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\tan"}},{"type":"text","text":" at "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{6}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Reciprocal identities for "},{"type":"inlineMath","attrs":{"latex":"\\csc"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sec"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\cot"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the reference angle for "},{"type":"inlineMath","attrs":{"latex":"\\frac{5\\pi}{6}"}},{"type":"text","text":" and its location on the unit circle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\sin\\left(\\frac{5\\pi}{6}\\right)"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{5\\pi}{6}\\right)"}},{"type":"text","text":" using the unit circle."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Calculate "},{"type":"inlineMath","attrs":{"latex":"\\tan\\left(\\frac{5\\pi}{6}\\right)"}},{"type":"text","text":" as "},{"type":"inlineMath","attrs":{"latex":"\\frac{\\sin\\left(\\frac{5\\pi}{6}\\right)}{\\cos\\left(\\frac{5\\pi}{6}\\right)}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find "},{"type":"inlineMath","attrs":{"latex":"\\csc"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"\\sec"}},{"type":"text","text":", and "},{"type":"inlineMath","attrs":{"latex":"\\cot"}},{"type":"text","text":" using reciprocal identities."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q6. 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For "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2 \\sin(3x)"}},{"type":"text","marks":[{"type":"bold"}],"text":", find the period, amplitude, and intercepts. Sketch the graph on the interval "},{"type":"inlineMath","attrs":{"latex":"[-\\pi, \\pi]"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Sine Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your understanding of how to analyze and graph sine functions with amplitude and period changes, and how to find intercepts."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Amplitude: "},{"type":"inlineMath","attrs":{"latex":"|A|"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"f(x) = A \\sin(Bx)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Period: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Intercepts: Solve "},{"type":"inlineMath","attrs":{"latex":"f(x) = 0"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts; "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the amplitude "},{"type":"inlineMath","attrs":{"latex":"|A|"}},{"type":"text","text":" and period "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2 \\sin(3x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts by solving "},{"type":"inlineMath","attrs":{"latex":"-2 \\sin(3x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept by evaluating "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph over "},{"type":"inlineMath","attrs":{"latex":"[-\\pi, \\pi]"}},{"type":"text","text":" using the amplitude, period, and intercepts."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q12. For "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\cos\\left(\\frac{x}{2}\\right)"}},{"type":"text","marks":[{"type":"bold"}],"text":", find the period, amplitude, and intercepts. Sketch the graph on the interval "},{"type":"inlineMath","attrs":{"latex":"[-5\\pi, 5\\pi]"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Cosine Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze and graph cosine functions with period changes, and to find intercepts."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Amplitude: "},{"type":"inlineMath","attrs":{"latex":"|A|"}},{"type":"text","text":" where "},{"type":"inlineMath","attrs":{"latex":"f(x) = A \\cos(Bx)"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Period: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Intercepts: Solve "},{"type":"inlineMath","attrs":{"latex":"f(x) = 0"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts; "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the amplitude "},{"type":"inlineMath","attrs":{"latex":"|A|"}},{"type":"text","text":" and period "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\cos\\left(\\frac{x}{2}\\right)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts by solving "},{"type":"inlineMath","attrs":{"latex":"\\cos\\left(\\frac{x}{2}\\right) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept by evaluating "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph over "},{"type":"inlineMath","attrs":{"latex":"[-5\\pi, 5\\pi]"}},{"type":"text","text":" using the amplitude, period, and intercepts."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q13. State the 3 different (but equivalent) Pythagorean identities."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Pythagorean Trigonometric Identities"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your knowledge of the fundamental Pythagorean identities for trigonometric functions, which are essential for simplifying and solving trigonometric equations."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"\\sin^2 x + \\cos^2 x = 1"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide by "},{"type":"inlineMath","attrs":{"latex":"\\sin^2 x"}},{"type":"text","text":" or "},{"type":"inlineMath","attrs":{"latex":"\\cos^2 x"}},{"type":"text","text":" to get the other two identities."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Write the basic identity: "},{"type":"inlineMath","attrs":{"latex":"\\sin^2 x + \\cos^2 x = 1"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\sin^2 x"}},{"type":"text","text":" to get the second identity."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Divide both sides by "},{"type":"inlineMath","attrs":{"latex":"\\cos^2 x"}},{"type":"text","text":" to get the third identity."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q14. For "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2 \\cos(3x)"}},{"type":"text","marks":[{"type":"bold"}],"text":", find the period, amplitude, and intercepts. Sketch the graph on the interval "},{"type":"inlineMath","attrs":{"latex":"\\left[-\\frac{5\\pi}{6}, \\frac{5\\pi}{6}\\right]"}},{"type":"text","marks":[{"type":"bold"}],"text":"."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Cosine Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to analyze and graph cosine functions with amplitude and period changes, and to find intercepts."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Amplitude: "},{"type":"inlineMath","attrs":{"latex":"|A|"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Period: "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Intercepts: Solve "},{"type":"inlineMath","attrs":{"latex":"f(x) = 0"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts; "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept"}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Identify the amplitude "},{"type":"inlineMath","attrs":{"latex":"|A|"}},{"type":"text","text":" and period "},{"type":"inlineMath","attrs":{"latex":"\\frac{2\\pi}{|B|}"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2 \\cos(3x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercepts by solving "},{"type":"inlineMath","attrs":{"latex":"-2 \\cos(3x) = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"-intercept by evaluating "},{"type":"inlineMath","attrs":{"latex":"f(0)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph over "},{"type":"inlineMath","attrs":{"latex":"\\left[-\\frac{5\\pi}{6}, \\frac{5\\pi}{6}\\right]"}},{"type":"text","text":" using the amplitude, period, and intercepts."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q15. Sketch the graph of "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{\\pi} \\tan^{-1} x"}},{"type":"text","marks":[{"type":"bold"}],"text":", explicitly showing all intercepts and horizontal asymptotes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Graphing Inverse Trigonometric Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to graph the arctangent function, including scaling and identifying key features such as intercepts and asymptotes."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"f(x) = \\frac{2}{\\pi} \\tan^{-1} x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Horizontal asymptotes for "},{"type":"inlineMath","attrs":{"latex":"\\tan^{-1} x"}},{"type":"text","text":" are at "},{"type":"inlineMath","attrs":{"latex":"y = \\pm 1"}},{"type":"text","text":" after scaling."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercept at "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":"."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Recall that "},{"type":"inlineMath","attrs":{"latex":"\\tan^{-1} x"}},{"type":"text","text":" has horizontal asymptotes at "},{"type":"inlineMath","attrs":{"latex":"y = \\frac{\\pi}{2}"}},{"type":"text","text":" and "},{"type":"inlineMath","attrs":{"latex":"y = -\\frac{\\pi}{2}"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Multiply by "},{"type":"inlineMath","attrs":{"latex":"\\frac{2}{\\pi}"}},{"type":"text","text":" to scale the range to "},{"type":"inlineMath","attrs":{"latex":"(-1, 1)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Find the "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":"-intercept by setting "},{"type":"inlineMath","attrs":{"latex":"x = 0"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Sketch the graph, showing the intercept and asymptotes."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q16. For the following functions "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":", find "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","marks":[{"type":"bold"}],"text":" and state the range of "},{"type":"inlineMath","attrs":{"latex":"f"}},{"type":"text","marks":[{"type":"bold"}],"text":" in interval notation."}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(a) "},{"type":"inlineMath","attrs":{"latex":"f(x) = -2 \\cos(3x)"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x \\in [0, \\frac{\\pi}{3}]"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(b) "},{"type":"inlineMath","attrs":{"latex":"f(x) = 2 \\sin x - 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x \\in [ -\\frac{\\pi}{2}, \\frac{\\pi}{2} ]"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(c) "},{"type":"inlineMath","attrs":{"latex":"f(x) = \\tan\\left(\\frac{x}{2}\\right) + 1"}},{"type":"text","text":", "},{"type":"inlineMath","attrs":{"latex":"x \\in (-\\pi, \\pi)"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Inverse Functions and Ranges"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to find the inverse of trigonometric functions and to determine the range of the original function."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"To find "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":", solve "},{"type":"inlineMath","attrs":{"latex":"y = f(x)"}},{"type":"text","text":" for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Range: Set of all possible output values for "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":" over the given domain."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each function, set "},{"type":"inlineMath","attrs":{"latex":"y = f(x)"}},{"type":"text","text":" and solve for "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" in terms of "},{"type":"inlineMath","attrs":{"latex":"y"}},{"type":"text","text":" to find "},{"type":"inlineMath","attrs":{"latex":"f^{-1}(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Use the domain of "},{"type":"inlineMath","attrs":{"latex":"x"}},{"type":"text","text":" to determine the range of "},{"type":"inlineMath","attrs":{"latex":"f(x)"}},{"type":"text","text":"."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Express the range in interval notation."}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Try solving on your own before revealing the answer!"}]},{"type":"heading","attrs":{"textAlign":null,"level":3},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Q17. Solve the following equations:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(a) "},{"type":"inlineMath","attrs":{"latex":"3 \\sin^{-1}(7x) = -\\pi"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(b) "},{"type":"inlineMath","attrs":{"latex":"3 \\tan^{-1} x = 4\\pi"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(c) "},{"type":"inlineMath","attrs":{"latex":"4 \\cos^{-1} x - 2\\pi = 2 \\cos^{-1} x"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"(d) "},{"type":"inlineMath","attrs":{"latex":"6 \\cos^{-1}(2x) = -\\pi"}}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Background"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Topic: Solving Equations Involving Inverse Trigonometric Functions"}]},{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"This question tests your ability to solve equations involving inverse trigonometric functions by isolating the variable and applying the definitions of the inverse functions."}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"underline"}],"text":"Key Terms and Formulas:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Inverse trigonometric function definitions:"}]},{"type":"bulletList","content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y = \\sin^{-1} x \\implies x = \\sin y"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y = \\cos^{-1} x \\implies x = \\cos y"}}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"inlineMath","attrs":{"latex":"y = \\tan^{-1} x \\implies x = \\tan y"}}]}]}]}]}]},{"type":"heading","attrs":{"textAlign":null,"level":4},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Step-by-Step Guidance"}]},{"type":"orderedList","attrs":{"start":1,"type":null},"content":[{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"For each equation, isolate the inverse trigonometric function."}]}]},{"type":"listItem","content":[{"type":"paragraph","attrs":{"textAlign":null},"content":[{"type":"text","text":"Appl