Section 5.5 (part 1, pages 411-413): Double-Angle Formulas

SINE

COSINE

TANGENT

*These formulas are the most commonly used multiple-angle formulas. Like other trigonometric identities, they are used to rewrite trigonometric functions in more convenient forms.

Proof of the SINE formula:

Proof of the COSINE formula:

Proof of the TANGENT formula:

*Compared to others, the proofs for the double-angle formulas are simple. Their basis are the Sum & Difference Formulas for sine, cosine, and tangent (Section 5.4, page 404).

Practice Problem (page 411): Solving a Double-Angle Equation

Find all solutions of 2cosx + sin2x = 0

1. 2cosx + 2sinxcosx = 0 (substitute the SINE double-angle formula for sin2x)

2. 2cosx(1 + sinx) = 0 (factor out a 2cosx)

3. 2cosx = 0, 1 + sinx = 0 (set both factors equal to 0)

4. cosx = 0, sin = -1 (simplify both factors)

5. x = , x = (use prior knowledge of the unit circle to find all solutions for x on the interval )

6. x = (eliminate duplicate solutions)

*Depending on the period of the equation, there may be additional answers.