Precalculus Trigonometry: Identities, Equations, and Exact Values

Trigonometric Functions and Identities

Basic Trigonometric Functions

Trigonometric functions relate the angles of a triangle to the lengths of its sides. The primary functions are sine, cosine, and tangent, commonly abbreviated as sin, cos, and tan.

• Sine (sin θ): Ratio of the length of the side opposite angle θ to the hypotenuse.

• Cosine (cos θ): Ratio of the length of the adjacent side to the hypotenuse.

• Tangent (tan θ): Ratio of the length of the opposite side to the adjacent side.

Example: If $\sin \theta = \frac{4}{5}$ and $\theta$ is in the first quadrant, then $\cos \theta = \frac{3}{5}$ and $\tan \theta = \frac{4}{3}$.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship among the trigonometric functions:

• $\sin^2 x + \cos^2 x = 1$

This identity is used to find one trigonometric value given another.

Reciprocal and Quotient Identities

• $\sec x = \frac{1}{\cos x}$

• $\csc x = \frac{1}{\sin x}$

• $\cot x = \frac{1}{\tan x}$

• $\tan x = \frac{\sin x}{\cos x}$

Solving Trigonometric Equations

Solving for Angles

To solve equations such as $\sin x = k$ or $\cos x = k$, use the inverse trigonometric functions and consider the periodicity of the functions.

• For $\sin x = k$, solutions are $x = \arcsin k + 2\pi n$ and $x = \pi - \arcsin k + 2\pi n$, where $n$ is any integer.

• For $\cos x = k$, solutions are $x = \arccos k + 2\pi n$ and $x = -\arccos k + 2\pi n$.

Example: $\sin x = -\frac{1}{2}$ has solutions $x = \frac{7\pi}{6} + 2\pi n$ and $x = \frac{11\pi}{6} + 2\pi n$.

Quadratic Trigonometric Equations

Equations such as $(\tan x + 1)(2 \sin x - 1) = 0$ can be solved by setting each factor to zero and solving for $x$.

• For $\tan x + 1 = 0$, $\tan x = -1 \implies x = -\frac{\pi}{4} + \pi n$

• For $2 \sin x - 1 = 0$, $\sin x = \frac{1}{2} \implies x = \frac{\pi}{6} + 2\pi n$ or $x = \frac{5\pi}{6} + 2\pi n$

Trigonometric Identities and Exact Values

Sum and Difference Formulas

These formulas allow the calculation of trigonometric functions of sums or differences of angles:

• $\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b$

• $\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b$

• $\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

Example: $\cos\left(\frac{\pi}{2} - x\right) = \sin x$

Double Angle and Half Angle Formulas

• $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

• $\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}$

Example: If $\cos \theta = \frac{8}{10}$ and $\theta$ is in quadrant IV, then $\sin 2\theta$ and $\cos 2\theta$ can be found using the double angle formulas.

Inverse Trigonometric Functions

Definitions and Properties

Inverse trigonometric functions return the angle whose trigonometric function equals a given value.

• $y = \arcsin x$ means $\sin y = x$ and $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• $y = \arccos x$ means $\cos y = x$ and $0 \leq y \leq \pi$

• $y = \arctan x$ means $\tan y = x$ and $-\frac{\pi}{2} < y < \frac{\pi}{2}$

Example: $\arccos \frac{1}{2} = \frac{\pi}{3}$

Solving Trigonometric Equations in Intervals

Finding All Solutions

To find all solutions in a given interval, use the periodicity of the trigonometric functions and the principal values from the inverse functions.

• For $\sin x = 0.7$ in $[0, 360^\circ)$, use $x = \arcsin 0.7$ and $x = 180^\circ - \arcsin 0.7$.

Applications and Problem Solving

Using Sketches and Reference Triangles

Reference triangles and sketches help determine the sign and value of trigonometric functions in different quadrants.

• Given $\cos \theta = \frac{8}{10}$ in quadrant IV, $\sin \theta$ is negative and $\tan \theta$ is negative.

Evaluating Trigonometric Expressions

Use identities and known values to evaluate expressions such as $\sin(45^\circ + 180^\circ)$ or $\cos\left(\frac{\pi}{2} - x\right)$.

Summary Table: Key Trigonometric Identities

Identity	Formula
Pythagorean	$\sin^2 x + \cos^2 x = 1$
Sum Formula (Sine)	$\sin(a + b) = \sin a \cos b + \cos a \sin b$
Sum Formula (Cosine)	$\cos(a + b) = \cos a \cos b - \sin a \sin b$
Double Angle (Sine)	$\sin 2x = 2 \sin x \cos x$
Double Angle (Cosine)	$\cos 2x = \cos^2 x - \sin^2 x$
Reciprocal (Secant)	$\sec x = \frac{1}{\cos x}$
Quotient (Tangent)	$\tan x = \frac{\sin x}{\cos x}$

Additional info:

• Some problems require using the unit circle and knowledge of reference angles.

• Solving trigonometric equations often involves considering all possible solutions within a specified interval due to the periodic nature of the functions.

• Inverse trigonometric functions are used to find exact values of angles given a trigonometric ratio.