6.2: Angle Sum and Double-Angle Identities | Precalculus

How it works

Angle-sum formulas give exact values by combining familiar angles. A double-angle formula is a special case. Substitution should keep the subtraction signs and the quadrant signs of every sine and cosine.

\sin(a+b)=\sin a\cos b+\cos a\sin b,\quad\cos(2a)=1-2\sin^2a

Worked examples

Example 1

Find $\displaystyle \sin(\pi/12)$ exactly.

Use $\displaystyle \pi/12=\pi/4-\pi/6$ . Then $\displaystyle \sin(\pi/12)=(\sqrt{6}-\sqrt{2})/4$ .

Example 2

If $\displaystyle \sin x=3/5$ and x is acute, find $\displaystyle \sin(2x)$ .

Cosine is $\displaystyle 4/5$ . Thus $\displaystyle \sin(2x)=2(3/5)(4/5)=24/25$ .

Problems

Problem 6.2.1

Find $\displaystyle \cos(\pi/12)$ exactly.

Show worked solution

Use the difference formula: $\displaystyle \cos(\pi/4-\pi/6)=\sqrt{6}/4+\sqrt{2}/4$ .

Problem 6.2.2

If $\displaystyle \sin x=3/5$ , find $\displaystyle \cos(2x)$ .

Show worked solution

$\displaystyle 1-2\sin^2x=1-18/25=7/25$ , regardless of which allowed quadrant contains x.

Problem 6.2.3

Explain why $\displaystyle \sin(a+b)=\sin a+\sin b$ is false.

Show worked solution

Take $\displaystyle a=b=\pi/2$ . The left side is $\displaystyle 0$ , but the right side is $\displaystyle 2$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 6.2.1

Problem 6.2.2

Problem 6.2.3

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