Unit 4 Test Review: Trigonometric Functions – Study Notes

Trigonometric Functions

Angles in Standard Position

Angles in standard position have their vertex at the origin and their initial side along the positive x-axis. The terminal side is determined by the direction and magnitude of rotation.

• Reference Angle: The smallest angle between the terminal side and the x-axis.

• Degree and Radian Measure: To convert between degrees and radians, use the formula:

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$ $\text{Degrees} = \text{Radians} \times \frac{180}{\pi}$

• Example: The point $(-1, \sqrt{3})$ is on the terminal side of an angle. Find the angle in degrees and radians.

Additional info: The angle is in the second quadrant. Use the arctangent function and reference triangles to find the angle.

Evaluating Trigonometric Functions Without a Calculator

Common angles (such as $0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$) have known sine, cosine, and tangent values. Inverse trigonometric functions return the angle whose trigonometric value is given.

• Key Values:

Angle	$\sin$	$\cos$	$\tan$
$0$	$0$	$1$	$0$
$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$\frac{\pi}{2}$	$1$	$0$	undefined

• Example: $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$

Right Triangle Trigonometry

For a right triangle with sides $a$, $b$, and hypotenuse $c$:

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

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

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

Example: In $\triangle ABC$ with $AB = 12$, $BC = 5$, find all six trigonometric functions for $\angle A$.

Additional info: Use the Pythagorean theorem to find the missing side, then compute sine, cosine, tangent, and their reciprocals.

Solving Right Triangles

Given two sides or one side and one angle (other than the right angle), use trigonometric ratios to find missing sides or angles.

• Example: In a right triangle, $B = 72^\circ$, $b = 24$ cm. Find the other sides using sine and cosine laws.

Applications of Trigonometry

Trigonometry is used to solve real-world problems involving angles of elevation and depression, and lengths of sides in right triangles.

• Example: A 200 ft guy wire makes a $55^\circ$ angle with the ground. The height of the tower is $200 \sin 55^\circ$.

Graphs of Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent have characteristic graphs with specific periods, amplitudes, and (for tangent, secant, cosecant, cotangent) asymptotes.

• Period: The length of one complete cycle.

• Amplitude: The maximum value from the midline (for sine and cosine).

• Asymptotes: Lines where the function is undefined (for tangent, cotangent, secant, cosecant).

Function	Period	Amplitude	Asymptotes
$y = \sin x$	$2\pi$	$1$	None
$y = \cos x$	$2\pi$	$1$	None
$y = \tan x$	$\pi$	None	$x = \frac{\pi}{2} + n\pi$
$y = \sec x$	$2\pi$	None	$x = \frac{\pi}{2} + n\pi$

• Example: $y = \sin(x + \frac{\pi}{2})$ has period $2\pi$, amplitude $1$.

Domain and Range of Trigonometric Functions

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: For $y = -3\cos(x + \frac{\pi}{2}) + 2$, the range is $[-1, 1]$ scaled and shifted to $[-1 \times 3 + 2, 1 \times 3 + 2] = [-1, 1]$.

Inverse Trigonometric Functions

Inverse trigonometric functions return the angle whose trigonometric value is given. Their principal values are restricted to specific intervals.

• $\arcsin x$ or $\sin^{-1} x$: $-1 \leq x \leq 1$, returns $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• $\arccos x$ or $\cos^{-1} x$: $-1 \leq x \leq 1$, returns $0 \leq y \leq \pi$

• $\arctan x$ or $\tan^{-1} x$: $x \in \mathbb{R}$, returns $-\frac{\pi}{2} < y < \frac{\pi}{2}$

• Example: $\arcsin\left(\frac{1}{2}\right) = \frac{\pi}{6}$

Writing Trigonometric Equations

General form for sine and cosine functions:

$y = A \sin(B(x - C)) + D$ $y = A \cos(B(x - C)) + D$

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Phase Shift: $C$

• Vertical Shift: $D$

• Example: A cosine function with amplitude 6, period $\pi$, and phase shift $\frac{\pi}{6}$ to the left: $y = 6\cos\left(2(x + \frac{\pi}{6})\right)$

End Behavior of Trigonometric Functions

For rational trigonometric functions, analyze the limits as $x \to \infty$ or $x \to -\infty$ to determine end behavior.

• Example: For $f(x) = \frac{\sin x}{x}$, as $x \to \infty$, $f(x) \to 0$.

Summary Table: Six Trigonometric Functions

Function	Definition
$\sin \theta$	$\frac{\text{opposite}}{\text{hypotenuse}}$
$\cos \theta$	$\frac{\text{adjacent}}{\text{hypotenuse}}$
$\tan \theta$	$\frac{\text{opposite}}{\text{adjacent}}$
$\csc \theta$	$\frac{\text{hypotenuse}}{\text{opposite}}$
$\sec \theta$	$\frac{\text{hypotenuse}}{\text{adjacent}}$
$\cot \theta$	$\frac{\text{adjacent}}{\text{opposite}}$