Trigonometric Functions
GradesAlgebra II & TrigonometryTrigonometric FunctionsSum & Difference Identities

Sum & Difference Identities

🌊 Trigonometric Functions
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Sum & Difference Identities

Evaluate trig functions at non-standard angles using sum and difference formulas.

The sum and difference identities express sin(A ± B) and cos(A ± B) in terms of trig functions of A and B separately. They allow exact evaluation at angles like 75° = 45° + 30°, which are not on the unit circle directly.

sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A\cos B + \cos A\sin B
sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A\cos B - \cos A\sin B
cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos(A + B) = \cos A\cos B - \sin A\sin B
cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A\cos B + \sin A\sin B
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Memory trick: Sine formulas match the sign (sin(A+B) has +, sin(A−B) has −). Cosine formulas are opposite to the sign (cos(A+B) has −, cos(A−B) has +).

✏️Exact Value of sin(75°)
sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4.
sin⁡75∘=6+24\sin 75^\circ = \frac{\sqrt{6}+\sqrt{2}}{4}
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Remember This!

The double-angle formulas are special cases: sin(2A) = sin(A + A) and cos(2A) = cos(A + A). You can re-derive double-angle identities from sum identities.

✏️ Try It!

Using the difference identity, what is the exact value of cos(15°)?

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