Precalculus Review: Trigonometric Functions, Identities, and Applications

Trigonometric Functions and Their Properties

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find angles when given the value of a trigonometric function. The principal values are typically restricted to ensure the function is one-to-one.

• arcsin(x): The angle whose sine is x, with range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

• arccos(x): The angle whose cosine is x, with range $[0, \pi]$.

• arctan(x): The angle whose tangent is x, with range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

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

Evaluating Trigonometric Expressions

To evaluate expressions such as $\sin\left(\arccos\left(-\frac{3}{5}\right)\right)$, use right triangle relationships or the Pythagorean identity.

• Let $\theta = \arccos\left(-\frac{3}{5}\right)$. Then $\cos\theta = -\frac{3}{5}$.

• Find $\sin\theta$ using $\sin^2\theta + \cos^2\theta = 1$.

Example: $\sin\theta = \sqrt{1 - \left(-\frac{3}{5}\right)^2} = \frac{4}{5}$ (sign depends on quadrant).

Exact Values in Degrees

Some trigonometric values are commonly memorized for special angles (e.g., $30^\circ$, $45^\circ$, $60^\circ$).

• $\sin(30^\circ) = \frac{1}{2}$

• $\cos(60^\circ) = \frac{1}{2}$

• $\tan(45^\circ) = 1$

Trigonometric Identities and Simplification

Basic Trigonometric Identities

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$

• Quotient Identities: $\tan\theta = \frac{\sin\theta}{\cos\theta}$, $\cot\theta = \frac{\cos\theta}{\sin\theta}$

• Reciprocal Identities: $\csc\theta = \frac{1}{\sin\theta}$, $\sec\theta = \frac{1}{\cos\theta}$

Using Identities to Find Other Trig Functions

If one trigonometric function value is known and the quadrant is specified, use identities to find the remaining functions.

• Given $\sin\theta = x$, $\cos\theta = \pm\sqrt{1 - x^2}$ (sign depends on quadrant).

Simplifying and Proving Identities

To prove an identity, manipulate one or both sides using algebraic and trigonometric identities until both sides are equal.

• Example: Prove $\tan x \tan y = \sec y - \cos y$ (requires algebraic manipulation and use of identities).

Even and Odd Functions

Definitions

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain.

Examples: $\cos(x)$ is even, $\sin(x)$ is odd.

Factoring Trigonometric Expressions

Factoring is used to simplify expressions or solve equations. Look for common factors or use identities to rewrite expressions.

• Example: $2\sin x + 3\sin x = \sin x (2 + 3) = 5\sin x$

Solving Right Triangles

Right Triangle Relationships

• Use the Pythagorean theorem: $a^2 + b^2 = c^2$

• Trigonometric ratios: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Example: Given $a = 7$, $c = 14$, find $b$ using $b = \sqrt{c^2 - a^2}$.

Applications of Trigonometry

Angles of Elevation and Depression

These concepts are used to solve real-world problems involving heights and distances.

• Angle of Elevation: The angle above horizontal from the observer's eye to an object.

• Angle of Depression: The angle below horizontal from the observer's eye to an object.

Example: If a 150-foot tall building casts a shadow of 200 feet, the angle of elevation $\theta$ is found using $\tan\theta = \frac{150}{200}$.

Degree and Radian Measure

Conversion Between Degrees and Radians

• To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

• To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Example: $45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4}$ radians.

Summary Table: Common Trigonometric Values

Angle (Degrees)	Angle (Radians)	sin	cos	tan
0	0	0	1	0
30	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1
60	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90	$\frac{\pi}{2}$	1	0	undefined

Additional info: This table summarizes the most commonly used trigonometric values for special angles, which are essential for solving many precalculus problems.