Sum, Difference & Double-Angle Identities (College Board AP® Precalculus): Study Guide

Sum identities

What are the sum identities for sine and cosine?

• The sum identity for sine expresses $\sin (\alpha +\beta )$ in terms of the sine and cosine of $\alpha$ and $\beta$ separately

  • $\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

• The sum identity for cosine expresses $\cos (\alpha +\beta )$ similarly

  • $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

• These identities hold for all values of $\alpha$ and $\beta$

How are sum identities used?

• One common use is to find the exact value of a trigonometric function at a non-standard angle by writing that angle as a sum of two standard angles

  • E.g. $\frac{7\pi }{12}=\frac{\pi }{3}+\frac{\pi }{4}$

    • so $\sin \frac{7\pi }{12}$ can be found using the sum identity with $\alpha =\frac{\pi }{3}$ and $\beta =\frac{\pi }{4}$

• Another common use is to rewrite trigonometric expressions in equivalent forms

  • This can be helpful when solving equations or simplifying

Difference identities

What are the difference identities?

• The difference identities are obtained from the sum identities

  • by replacing $\beta$ with $-\beta$

    • and using the fact that sine is odd ($\sin (-\beta )=-\sin \beta$)

    • and cosine is even ($\cos (-\beta )=\cos \beta$):

  • Doing this gives you

    • $\sin (\alpha -\beta )=\sin \alpha \cos \beta -\cos \alpha \sin \beta$

    • $\cos (\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta$

• Notice the sign patterns

  • The sine difference identity has a minus sign between its two terms

    • whereas the sine sum identity has a plus

  • The cosine difference identity has a plus sign between its two terms

    • whereas the cosine sum identity has a minus

How are difference identities used?

• Like sum identities, difference identities can be used to find exact values at non-standard angles by writing that angle as a difference of two standard angles

  • E.g. $\frac{\pi }{12}=\frac{\pi }{3}-\frac{\pi }{4}$, so $\sin \frac{\pi }{12}$ can be found using the difference identity

Double-angle identities

What are the double-angle identities?

• The double-angle identities are obtained from the sum identities by setting $\alpha =\beta =\theta$:

  • For sine

    • $\sin \left(2\theta \right)=\sin (\theta +\theta )=\sin \theta \cos \theta +\cos \theta \sin \theta =2\sin \theta \cos \theta$

  • For cosine

    • $\cos \left(2\theta \right)=\cos (\theta +\theta )=\cos \theta \cos \theta -\sin \theta \sin \theta ={\cos }^2\theta -{\sin }^2\theta$

• This gives the two double-angle identities

$\mathbf{sin}\mathbf{\left(}\mathbf{2}\mathit{\theta }\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{sin}\mathit{\theta }\mathbf{cos}\mathit{\theta }$

$\mathbf{cos}\mathbf{\left(}\mathbf{2}\mathit{\theta }\mathbf{\right)}\mathbf{=}{\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }\mathbf{-}{\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }$

What are the alternative forms of the cosine double-angle identity?

• The cosine double-angle identity can be rewritten in two alternative forms by using the Pythagorean identity ${\sin }^2\theta +{\cos }^2\theta =1$:

  • Substituting ${\sin }^2\theta =1-{\cos }^2\theta$ gives

    • $\cos \left(2\theta \right)={\cos }^2\theta -(1-{\cos }^2\theta )=2{\cos }^2\theta -1$

  • Substituting ${\cos }^2\theta =1-{\sin }^2\theta$ gives

    • $\cos \left(2\theta \right)=(1-{\sin }^2\theta )-{\sin }^2\theta =1-2{\sin }^2\theta$

• All three forms are equivalent

$\mathbf{cos}\mathbf{\left(}\mathbf{2}\mathit{\theta }\mathbf{\right)}\mathbf{=}{\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }\mathbf{-}{\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }\mathbf{=}\mathbf{2}{\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{1}\mathbf{-}\mathbf{2}{\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }$

• The different forms are useful in different situations

  • e.g. if an expression already contains ${\sin }^2\theta$, then the $1-2{\sin }^2\theta$ form may be most convenient

How are double-angle identities used?

• They are commonly used to rewrite expressions involving $\sin \left(2x\right)$ or $\cos \left(2x\right)$ in terms of $\sin x$ and $\cos x$

  • or vice versa

• They also arise when solving equations that mix single-angle and double-angle terms

  • Applying the double-angle identity can convert the equation to one involving only single-angle terms