Sum and Difference Identities

Sum and difference identities express trig values of combined angles using the values of the individual angles.

Interpret → represent → solve → check. The problem is translated into a mathematical representation, solved with an appropriate operation, and checked against the original quantities and conditions.

Trigonometric functions do not distribute over angle addition; their correct combination rules mix sine and cosine.

Sine identities

For any angles $A$ and $B$,

\sin(A+B)=\sin A\cos B+\cos A\sin B,

\sin(A-B)=\sin A\cos B-\cos A\sin B.

For sine, the sign in the formula matches the sign between the angles.

Worked example: exact value of cosine 75°

Worked example: exact sine of 15°

Write $15^\circ=45^\circ-30^\circ$:

\sin15^\circ =\sin45^\circ\cos30^\circ-\cos45^\circ\sin30^\circ =\frac{\sqrt6-\sqrt2}{4}.

Different angle decompositions can work, but choose special angles with known exact values.

Formula memory and checks

For sine: “same sign.” For cosine: “opposite sign.” Tangent follows sine-like signs in the numerator and cosine-like opposite signs in the denominator.

Use benchmark estimates, quadrant signs, or a calculator decimal to check an exact result after the algebra is complete.

Common mistakes

Writing $\sin(A+B)=\sin A+\sin B$. Trig functions do not distribute over addition.

Using the wrong cosine sign. Cosine switches the operation sign.

Losing radicals or denominators. Substitute exact values with parentheses.

Choosing angles without known exact values. Decompose strategically.

Ignoring the compound angle's quadrant. Use it to check the final sign.

Quick self-check

Is the compound angle written as a useful sum or difference?
Did I choose the sine, cosine, or tangent formula accurately?
Are special-angle values exact?
Does the final sign match the compound angle's quadrant?
Can a decimal estimate confirm the magnitude without replacing the exact answer?