Trigonometry Exam 3 Study Guide: Identities, Equations, and Applications

Trigonometric Identities and Equations

Proving Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables where both sides are defined. Proving identities is a fundamental skill in trigonometry, requiring manipulation of expressions using known formulas.

• Sum and Difference Formulas: These allow the calculation of trigonometric functions for sums or differences of angles.

• Double-Angle and Half-Angle Formulas: Useful for simplifying expressions and solving equations.

• Reciprocal and Pythagorean Identities: These relate the basic trigonometric functions to each other.

Example: Prove $\cos^2 x - \sin^2 x = \cos 2x$ using the double-angle formula for cosine.

Formula:

• $\cos 2x = \cos^2 x - \sin^2 x$

• $\sin 2x = 2\sin x \cos x$

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

Using Identities to Simplify Expressions

Trigonometric expressions can often be simplified by applying appropriate identities. This process is essential for solving equations and evaluating expressions.

• Sum-to-Product and Product-to-Sum Formulas: Convert sums into products and vice versa.

• Example: Simplify $\sin 75^\circ$ using sum and difference identities:

$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ$

Solving Trigonometric Equations

Exact Values Using Identities

Many trigonometric equations require finding the exact value of an expression using identities and known values.

• Example: Find $\sin(15^\circ)$ using the difference identity:

$\sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ$

Solving for Angles and Values

Given certain conditions (such as quadrant information or function values), you can solve for unknown angles or trigonometric values.

• Example: If $\sin \theta = \frac{3}{5}$ and $\theta$ is in Quadrant I, find $\cos \theta$ and $\tan \theta$.

Use the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$

$\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - (\frac{3}{5})^2} = \frac{4}{5}$

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3}{4}$

Applications of Trigonometric Identities

Double-Angle and Half-Angle Applications

Double-angle and half-angle identities are used to find values for angles that are multiples or fractions of known angles.

• Double-Angle: $\sin 2x$, $\cos 2x$, $\tan 2x$

• Half-Angle: $\sin \frac{x}{2}$, $\cos \frac{x}{2}$, $\tan \frac{x}{2}$

Example: Find $\sin 22.5^\circ$ using the half-angle identity:

$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$

Sum and Difference of Functions

Sum and difference formulas allow for the evaluation of trigonometric functions at non-standard angles.

• Example: $\tan(75^\circ) = \tan(45^\circ + 30^\circ)$

$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

Reference Table: Common Trigonometric Identities

Identity Type	Formula
Pythagorean	$\sin^2 x + \cos^2 x = 1$
Double-Angle	$\sin 2x = 2\sin x \cos x$ $\cos 2x = \cos^2 x - \sin^2 x$
Sum/Difference	$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$ $\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$
Half-Angle	$\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}$ $\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}$

Practice Problems Overview

• Prove and apply trigonometric identities.

• Use sum, difference, double-angle, and half-angle formulas to find exact values.

• Simplify expressions using identities.

• Solve equations involving trigonometric functions, given quadrant information.

• Apply identities to evaluate trigonometric functions at specific angles.

Additional info: The study guide covers topics from Chapters 5 and 6, focusing on trigonometric identities, equations, and their applications, which are central to college-level trigonometry.