10.5 Solving Trigonometric Equations

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📖 Main Content

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Section 1: The Six Trigonometric Functions — A Quick Recap

Before solving equations, you need to know all six trigonometric functions and how they relate to each other. Think of them as a family — three "main" functions and three "reciprocal" (flipped) functions.

The three main functions:

The three reciprocal functions (each is 1 divided by a main function):

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2.1 The Tangent Identity

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

This tells you that tan can always be written as sin divided by cos. Very useful when an equation mixes sin, cos, and tan.

2.2 The Pythagorean Identities

These three identities all come from the same source — the Pythagorean theorem (a² + b² = c²) applied to a unit circle.

Identity 1 (The most important one): $\sin^2 \theta + \cos^2 \theta = 1$

From this, you can rearrange to get:

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

Identity 2 (Involves sec and tan): $1 + \tan^2 \theta = \sec^2 \theta$

This comes from dividing Identity 1 through by $\cos^2 \theta$.

Rearranged: $\tan^2 \theta = \sec^2 \theta - 1$

Identity 3 (Involves cosec and cot):

$$\cot^2 \theta + 1 = \text{cosec}^2\ \theta$$

This comes from dividing Identity 1 through by $\sin^2 \theta$.

Rearranged: $\cot^2 \theta = \text{cosec}^2\ \theta - 1$