Precalculus Midterm Study Notes: Trigonometry, Functions, Logarithms, and More

Coterminal Angles and Reference Angles

Coterminal Angles

Coterminal angles are angles in standard position (with the same initial side) that share the same terminal side. They differ by integer multiples of $360^\circ$ or $2\pi$ radians.

• Finding Coterminal Angles: Add or subtract $360^\circ$ (or $2\pi$ radians) as many times as needed.

• Least Nonnegative Coterminal Angle: The smallest angle between $0$and $360^\circ$ (or $0$and $2\pi$) coterminal with the given angle.

• Formula: $\theta_{coterminal} = \theta + 360^\circ \times k$ or $\theta + 2\pi k$, where $k$ is any integer.

• Example: Find a coterminal angle for $450^\circ$: $450^\circ - 360^\circ = 90^\circ$ (least nonnegative coterminal angle)

Reference Angles

A reference angle is the acute angle formed by the terminal side of a given angle and the x-axis. It is always between $0$and$ (or $0$ and $\frac{\pi}{2}$ radians).

• Finding Reference Angle (Degrees):

  • Quadrant I: $\theta$

  • Quadrant II: $180^\circ - \theta$

  • Quadrant III: $\theta - 180^\circ$

  • Quadrant IV: $360^\circ - \theta$

• Finding Reference Angle (Radians):

  • Quadrant I: $\theta$

  • Quadrant II: $\pi - \theta$

  • Quadrant III: $\theta - \pi$

  • Quadrant IV: $2\pi - \theta$

• Example: Reference angle for $\frac{8\pi}{3}$:

  • First, find coterminal angle: $\frac{8\pi}{3} - 2\pi = \frac{8\pi}{3} - \frac{6\pi}{3} = \frac{2\pi}{3}$

  • Reference angle is $\frac{2\pi}{3}$ (Quadrant II): $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$

Trigonometric Functions and Identities

Evaluating Trigonometric Functions

Trigonometric functions can be evaluated using the unit circle, right triangles, or identities. The quadrant determines the sign of the function.

• Given tan $\theta$ and Quadrant: Use the sign of tangent and the quadrant to determine the signs of sine and cosine.

• Using Pythagorean Identities:

  • $\sin^2\theta + \cos^2\theta = 1$

  • $1 + \tan^2\theta = \sec^2\theta$

  • $1 + \cot^2\theta = \csc^2\theta$

• Example: If $\tan\theta = -3$ and $\theta$ is in Quadrant II, find $\sin\theta$ and $\cos\theta$.

  • Let $\tan\theta = \frac{\sin\theta}{\cos\theta} = -3$

  • In Quadrant II, $\sin\theta > 0$, $\cos\theta < 0$

  • Let $\sin\theta = 3k$, $\cos\theta = -k$

  • Use identity: $(3k)^2 + (-k)^2 = 1 \Rightarrow 9k^2 + k^2 = 1 \Rightarrow k^2 = \frac{1}{10}$

  • $\sin\theta = 3\left(\frac{1}{\sqrt{10}}\right) = \frac{3}{\sqrt{10}}$

  • $\cos\theta = -\frac{1}{\sqrt{10}}$

Simplifying Trigonometric Expressions

• Cofunction Identities: $\sin(90^\circ - \theta) = \cos\theta$, $\tan(90^\circ - \theta) = \cot\theta$, etc.

• Complementary Angles: Two angles are complementary if their sum is $90^\circ$ or $\frac{\pi}{2}$ radians.

• Example: $\sin(60^\circ) = \cos(30^\circ)$

Trigonometric Ratios and the Unit Circle

• Exact Values: Use the unit circle to find exact values for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$, etc.

• Example: $\sin(45^\circ) = \frac{\sqrt{2}}{2}$, $\cos(60^\circ) = \frac{1}{2}$

• Evaluating Cosecant: $\csc\theta = \frac{1}{\sin\theta}$

Acute Angles with Trig Ratios

• Finding Angle Given Ratio: Use inverse trigonometric functions, e.g., $\theta = \arcsin(0.5)$

• Special Angles: $30^\circ, 45^\circ, 60^\circ$ and their radian equivalents

Difference Quotient

Definition and Application

The difference quotient is used to compute the average rate of change of a function and is foundational for calculus.

• Formula: $\frac{f(x+h) - f(x)}{h}$

• For Square Root Functions: Rationalize the numerator to simplify.

• Example: For $f(x) = \sqrt{x}$:

  • $\frac{\sqrt{x+h} - \sqrt{x}}{h}$

  • Multiply numerator and denominator by $\sqrt{x+h} + \sqrt{x}$ to rationalize.

Right Triangle Trigonometry

Solving Right Triangles

Right triangle trigonometry relates the angles and sides of right triangles using sine, cosine, and tangent.

• Basic Ratios:

  • $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$

  • $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

  • $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

• Applications: Use trigonometric ratios to solve for unknown sides or angles in word problems.

• Example: Given $\theta = 30^\circ$ and hypotenuse $= 10$, find the opposite side: $\sin(30^\circ) = \frac{\text{opposite}}{10} \Rightarrow \text{opposite} = 10 \times 0.5 = 5$

Logarithmic Expressions and Equations

Expanding and Condensing Logarithms

Logarithmic properties allow us to expand or condense expressions for simplification or solving equations.

• Product Rule: $\log_b(MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Power Rule: $\log_b(M^k) = k \log_b M$

• Example (Expand): $\log_2(8x) = \log_2 8 + \log_2 x$

• Example (Condense): $2\log_3 x + \log_3 y = \log_3 x^2 + \log_3 y = \log_3(x^2y)$

Solving Logarithmic Equations

• Convert to Exponential Form: $\log_b a = c \Rightarrow a = b^c$

• Variable in Base: Use properties to isolate the variable, then exponentiate both sides if necessary.

• Example: $\log_x 16 = 2 \Rightarrow x^2 = 16 \Rightarrow x = 4$

Exponential Equations and Applications

Solving Exponential Equations

• Using Natural Logarithms: Take $\ln$ of both sides to bring down exponents.

• Base $e$: $e^{kx} = a \Rightarrow kx = \ln a \Rightarrow x = \frac{\ln a}{k}$

• Example: $2^x = 8 \Rightarrow x = 3$

Exponential Decay and Half-Life

• Exponential Decay Formula: $A = A_0 e^{-kt}$

• Half-Life Formula: $t_{1/2} = \frac{\ln 2}{k}$

• Example: If $A = 25$, $A_0 = 100$, $t = 10$, find $k$: $25 = 100 e^{-10k} \Rightarrow e^{-10k} = 0.25 \Rightarrow -10k = \ln 0.25 \Rightarrow k = -\frac{\ln 0.25}{10}$

Domain and Range

Finding Domain and Range

• From a Graph: Domain is the set of all $x$-values; range is the set of all $y$-values the function attains.

• Interval Notation: Use parentheses for open intervals, brackets for closed intervals. Example: $(-\infty, 2]$

• Example: For $f(x) = \sqrt{x-1}$, domain is $[1, \infty)$

Rational Functions

Key Properties

• Domain: All real numbers except where the denominator is zero.

• Removable Discontinuities: Occur where a factor cancels from numerator and denominator.

• Symmetry: Test by substituting $-x$ for $x$ (even/odd function tests).

• X and Y Intercepts:

  • X-intercept: Set numerator to zero.

  • Y-intercept: Set $x = 0$.

• Example: $f(x) = \frac{x-1}{x^2-1}$

  • Domain: $x \neq 1, -1$

  • Removable discontinuity at $x=1$ (factor cancels)

  • X-intercept: $x=1$

  • Y-intercept: $f(0) = \frac{-1}{-1} = 1$

Central Angles and Sectors

Central Angle and Area of a Sector

• Area of a Sector (Radians): $A = \frac{1}{2} r^2 \theta$

• Finding Central Angle: $\theta = \frac{2A}{r^2}$

• Example: If $A = 10$, $r = 4$, $\theta = \frac{2 \times 10}{16} = 1.25$ radians

Graphing and Transforming Functions

Function Transformations

• General Form: $y = a f(b(x-h)) + k$

• Transformations:

  • Horizontal Shift: $h$ units right if $h > 0$, left if $h < 0$

  • Vertical Shift: $k$ units up if $k > 0$, down if $k < 0$

  • Reflection: Over x-axis if $a < 0$, over y-axis if $b < 0$

  • Stretch/Compression: $|a| > 1$ stretches vertically, $0 < |a| < 1$ compresses

• Example: $y = -2f(3(x+1)) - 4$ is reflected over x-axis, vertically stretched by 2, horizontally compressed by 3, shifted left 1, and down 4.

Inverse Functions

Finding and Analyzing Inverses

• Finding Inverse: Swap $x$ and $y$ in $y = f(x)$, solve for $y$.

• Quadratic Functions: Restrict domain to ensure the function is one-to-one before finding the inverse.

• Domain and Range: The domain of $f$ becomes the range of $f^{-1}$ and vice versa.

• Example: $f(x) = x^2$, $x \geq 0$; inverse is $f^{-1}(x) = \sqrt{x}$

Inverse Functions from Graphs

• Domain and Range: Read from the graph; reflect over $y = x$ line.

• Evaluating Inverse: If $f(a) = b$, then $f^{-1}(b) = a$

Topic	Key Formula	Example
Coterminal Angles	$\theta + 360^\circ k$ or $\theta + 2\pi k$	$450^\circ$ coterminal: $90^\circ$
Difference Quotient	$\frac{f(x+h) - f(x)}{h}$	$f(x) = x^2 \Rightarrow 2x + h$
Logarithm Product Rule	$\log_b(MN) = \log_b M + \log_b N$	$\log_2(8x) = \log_2 8 + \log_2 x$
Area of Sector	$A = \frac{1}{2} r^2 \theta$	$r=4, \theta=1.25$; $A=10$
Exponential Decay	$A = A_0 e^{-kt}$	$A=25, A_0=100, t=10$

Additional info: Some examples and formulas were expanded for clarity and completeness. All topics are standard in a Precalculus curriculum.