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Sum and difference identities

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Sum and Difference Identities

The sum and difference identities give the sine, cosine, or tangent of an angle written as the sum or difference of two other angles, such as 75 degrees as 45 degrees plus 30 degrees. Learn the formulas for sine, cosine, and tangent, and how to use them to find exact trig values without a calculator.

What sum and difference identities do

The sum and difference identities let you find the sine, cosine, or tangent of an angle written as the sum or difference of two other angles — without needing a calculator. They break a single unfamiliar angle like 75° into two familiar ones, such as 45° + 30°.

Sum and difference identities Sine of a plus b equals sine a cosine b plus cosine a sine b; sine of a minus b equals sine a cosine b minus cosine a sine b. Cosine of a plus b equals cosine a cosine b minus sine a sine b; cosine of a minus b equals cosine a cosine b plus sine a sine b. Tangent of a plus or minus b equals tangent a plus or minus tangent b, over 1 minus or plus tangent a tangent b. Sum and difference identities sin(a ± b) = sin a cos b ± cos a sin b cos(a + b) = cos a cos b − sin a sin b cos(a − b) = cos a cos b + sin a sin b tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b)
The sum and difference identities for sine, cosine, and tangent.

The sine and cosine identities

For sine, the sum and difference identities share the same pattern: sin(a ± b) = sin a cos b ± cos a sin b. Cosine flips the sign: cos(a + b) subtracts (cos a cos b − sin a sin b), while cos(a − b) adds (cos a cos b + sin a sin b). Getting the signs right is the most common sticking point, so it helps to practice each one from a table like the one above.

The tangent identity

The tangent sum and difference identity comes from dividing the sine identity by the cosine identity: tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b). It uses the quotient identity for tangent as its starting point.

Worked example

Find sin 75° using sin(45° + 30°). Substitute known values: sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4. This exact value would be hard to find any other way.

Why this matters

Sum and difference identities are the foundation for the double-angle identities, which are just the special case where a and b are equal.

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