Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

Trigonometric Functions and Radian Measure

Definitions and Formulas

Trigonometric functions and radian measure are foundational concepts in precalculus, especially for understanding angles, circles, and periodic phenomena.

• Radian Measure: The measure of a central angle whose arc length is equal to the radius of the circle. One full revolution is $2\pi$ radians.

• Arc Length Formula: The length of an arc of a circle of radius $r$ subtended by a central angle $\theta$ (in radians) is given by: $s = r\theta$

• Area of a Sector: The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ (in radians): $A = \frac{1}{2} r^2 \theta$

• Conversion between Degrees and Radians: $1\ \text{radian} = \frac{180^\circ}{\pi}$ $1^\circ = \frac{\pi}{180}$ radians

Example: Find the radian measure of a central angle of a circle of radius 4 inches that intercepts an arc of length 3 inches. $\theta = \frac{s}{r} = \frac{3}{4}$ radians

Trigonometric Functions on the Unit Circle

Definitions and Evaluations

Trigonometric functions can be defined using the unit circle, where a point $P = (x, y)$ lies on the circle of radius $r$ at an angle $\theta$ from the positive x-axis.

• Sine: $\sin \theta = \frac{y}{r}$

• Cosine: $\cos \theta = \frac{x}{r}$

• Tangent: $\tan \theta = \frac{y}{x}$ (if $x \neq 0$)

• Cosecant: $\csc \theta = \frac{r}{y}$ (if $y \neq 0$)

• Secant: $\sec \theta = \frac{r}{x}$ (if $x \neq 0$)

• Cotangent: $\cot \theta = \frac{x}{y}$ (if $y \neq 0$)

Example: If $r = 17$ and $P = (x, y)$, then $\sin \theta = \frac{y}{17}$, $\cos \theta = \frac{x}{17}$, $\tan \theta = \frac{y}{x}$.

Trigonometric Identities

Fundamental Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides are defined.

Identity Type	Identities
Reciprocal	$\csc \theta = \frac{1}{\sin \theta}$ $\sec \theta = \frac{1}{\cos \theta}$ $\cot \theta = \frac{1}{\tan \theta}$
Quotient	$\tan \theta = \frac{\sin \theta}{\cos \theta}$ $\cot \theta = \frac{\cos \theta}{\sin \theta}$
Even/Odd	$\sin(-\theta) = -\sin \theta$ $\cos(-\theta) = \cos \theta$ $\tan(-\theta) = -\tan \theta$
Co-function	$\sin(\frac{\pi}{2} - \theta) = \cos \theta$ $\tan(\frac{\pi}{2} - \theta) = \cot \theta$ $\sec(\frac{\pi}{2} - \theta) = \csc \theta$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $1 + \cot^2 \theta = \csc^2 \theta$

Solving Right Triangles and Applications

Solving for Sides and Angles

Given a right triangle, trigonometric functions can be used to find unknown sides or angles.

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Solving for an angle: Use inverse trigonometric functions, e.g., $\theta = \arcsin(\frac{opposite}{hypotenuse})$

• Applications: Problems may involve finding distances, heights, or angles using trigonometric ratios and the Pythagorean theorem.

Example: In a triangle with sides 7 and 25, and hypotenuse 25, $\sin A = \frac{7}{25}$, $\tan A = \frac{7}{24}$.

Graphs of Trigonometric Functions

Basic Properties and Transformations

Trigonometric functions such as sine, cosine, and tangent have characteristic graphs with specific amplitude, period, and phase shift.

• Amplitude: The maximum value from the midline. For $y = a \sin x$, amplitude is $|a|$.

• Period: The length of one complete cycle. For $y = \sin(bx)$, period is $\frac{2\pi}{|b|}$.

• Phase Shift: Horizontal shift of the graph. For $y = \sin(x - c)$, phase shift is $c$ units right.

• Vertical Shift: For $y = \sin x + d$, the graph is shifted $d$ units up.

Example: $y = 3 \cos 5x$ has amplitude 3 and period $\frac{2\pi}{5}$.

Domain, Range, and Asymptotes

Function	Domain	Range	Zeros	Asymptotes
$y = \tan x$	$x \neq \frac{\pi}{2} + n\pi$	$(-\infty, \infty)$	$x = n\pi$	$x = \frac{\pi}{2} + n\pi$
$y = \cot x$	$x \neq n\pi$	$(-\infty, \infty)$	$x = \frac{\pi}{2} + n\pi$	$x = n\pi$
$y = \csc x$	$x \neq n\pi$	$(-\infty, -1] \cup [1, \infty)$	None	$x = n\pi$
$y = \sec x$	$x \neq \frac{\pi}{2} + n\pi$	$(-\infty, -1] \cup [1, \infty)$	None	$x = \frac{\pi}{2} + n\pi$

Inverse Trigonometric Functions

Definitions, Domains, and Ranges

Inverse trigonometric functions allow us to find angles given trigonometric ratios. Their domains and ranges are restricted to ensure they are functions.

Inverse Function	Domain	Range
Sine inverse ($\arcsin x$)	$[-1, 1]$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
Cosine inverse ($\arccos x$)	$[-1, 1]$	$[0, \pi]$
Tangent inverse ($\arctan x$)	$(-\infty, \infty)$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$
Cotangent inverse ($\arccot x$)	$(-\infty, \infty)$	$\left( 0, \pi \right)$

Simple Harmonic Motion

Equations and Applications

Simple harmonic motion describes periodic oscillations, such as springs or pendulums, and is modeled by sine or cosine functions.

• General Equation: $d = a \sin(\omega t)$ or $d = a \cos(\omega t)$, where $a$ is amplitude, $\omega$ is angular frequency.

• Frequency: Number of cycles per second, $f = \frac{\omega}{2\pi}$

• Period: Time for one cycle, $T = \frac{2\pi}{\omega}$

Example: For $d = 2 \cos 3\pi t$, amplitude is 2, period is $\frac{2}{3}$ seconds, frequency is $\frac{3}{2}$ oscillations/second.

Summary Table: Sine and Cosine Functions

	y = sin x	y = cos x
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[-1, 1]$	$[-1, 1]$
x-intercepts	$x = n\pi$	$x = \frac{\pi}{2} + n\pi$

Additional info: Some content, such as specific problem numbers and references to class notes, was omitted for clarity and replaced with general academic context and standard formulas.