 # Sine and Cosine Addition Formulas

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Examples, solutions, videos, worksheets, and activities to help PreCalculus students learn how to derive the sine and cosine addition formulas and how to use them to prove other trigonometric identities.

The following diagram shows the addition formulas for sine and cosine. Scroll down the page for more examples and solutions using the addition formulas. The cosine addition formula calculates the cosine of an angle that is either the sum or difference of two other angles. It arises from the law of cosines and the distance formula. By using the cosine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.

Proof of the trig identity, cosine of a sum formula: cos(a + b) = (cos a)(cos b) − (sin a)(sin b)
How to derive the cosine of a difference formula?
cos(a − b) = (cos a)(cos b) + (sin a)(sin b)

Starting with the cofunction identities, the sine addition formula is derived by applying the cosine difference formula. There are two main differences from the cosine formula: (1) the sine addition formula adds both terms, where the cosine addition formula subtracts and the subtraction formula adds; and (2) the sine formulas have sin-sin and cos-cos. Both formulas find values for angles.

How to derive the sine of a sum formula, sin(a + b) = (cos a)(sin b) + (sin a)(cos b) (using the cosine difference formula)?
Proof of the trig identity, sine of a sum formula: sin(a + b) = (cos a)(sin b) + (sin a)(cos b) The derivation of the sum and difference identities for cosine and sine
cos(a + b) = (cos a)(cos b) − (sin a)(sin b)
sin(a − b) = (sin a)(cos b) − (cos a)(sin b)
sin(a + b) = (sin a)(cos b) + (cos a)(sin b)
cos(a − b) = (cos a)(cos b) + (sin a)(sin b) Using the Sine and Cosine Addition Formulas to Prove Identities
Applying the cosine addition and sine addition formulas proves the cofunction, add pi, and supplementary angle identities.
Using the formulas, we see that
sin(π/2 − x) = cos(x), cos(π/2 − x) = sin(x);
that sin(x + π) = − sin(x), cos(x + π) = − cos(x);
and that sin(π − x) = sin(x), cos(π −x) = − cos(x).
The formulas also give the tangent of a difference formula, for tan(α − β).

How to use the sine and cosine addition formulas to prove the cofunction identities?

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