Precalculus

6.

Analytic Trigonometry

6.1.

Fundamental Trigonometric Identities

6.1.1.

Reciprocal Identities

6.1.2.

Quotient Identities

6.1.3.

Pythagorean Identities

6.1.3.1.

sin²θ + cos²θ = 1

6.1.3.2.

1 + tan²θ = sec²θ

6.1.3.3.

1 + cot²θ = csc²θ

6.1.4.

Even-Odd Identities

6.1.5.

Cofunction Identities

6.2.

Verifying Trigonometric Identities

6.2.1.

Strategies for Verification

6.2.1.1.

Work with One Side

6.2.1.2.

Convert to Sine and Cosine

6.2.1.3.

Factor and Simplify

6.2.1.4.

Multiply by Conjugates

6.2.2.

Common Techniques

6.2.2.1.

Combining Fractions

6.2.2.2.

Separating Fractions

6.2.2.3.

6.3.

Sum and Difference Formulas

6.3.1.

Sine Sum and Difference Formulas

6.3.2.

Cosine Sum and Difference Formulas

6.3.3.

Tangent Sum and Difference Formulas

6.3.4.

Applications of Sum and Difference Formulas

6.3.5.

Cofunction Identities from Sum Formulas

6.4.

Multiple-Angle Formulas

6.4.1.

Double-Angle Formulas

6.4.1.1.

Sine Double-Angle

6.4.1.2.

Cosine Double-Angle

6.4.1.3.

Tangent Double-Angle

6.4.2.

Power-Reducing Formulas

6.4.2.1.

Reducing Powers of Sine

6.4.2.2.

Reducing Powers of Cosine

6.4.2.3.

Reducing Powers of Tangent

6.4.3.

Half-Angle Formulas

6.4.3.1.

Sine Half-Angle

6.4.3.2.

Cosine Half-Angle

6.4.3.3.

Tangent Half-Angle

6.5.

Product-to-Sum and Sum-to-Product Formulas

6.5.1.

Product-to-Sum Formulas

6.5.2.

Sum-to-Product Formulas

6.5.3.

Applications in Simplification

6.6.

Solving Trigonometric Equations

6.6.1.

Basic Trigonometric Equations

6.6.1.1.

Linear in Trigonometric Functions

6.6.1.2.

Quadratic in Trigonometric Functions

6.6.2.

Using Fundamental Identities

6.6.3.

Using Sum and Difference Formulas

6.6.4.

Using Multiple-Angle Formulas

6.6.5.

Equations with Multiple Angles

6.6.6.

General Solutions

6.6.7.

Solutions in Given Intervals

6.6.8.

Checking Solutions

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7. Applications of Trigonometry