Angle sum and difference formulas: formula, calculator and worked example

What the symbols mean

$A$: the first angle
$B$: the second angle
$\sin (A + B)$: the sine of the sum A + B
$\cos (A + B)$: the cosine of the sum A + B
$∓$: "minus or plus": the opposite sign to the ± on the left

When to use it

Use them to find exact values for angles like 15°, 75° and 105° from the known values at 30°, 45° and 60°, to expand expressions such as $sin (x + 90°)$ , to combine $asin x + bcos x$ into a single sine wave, and to derive the double-angle formulas. Watch the cosine signs: the sum formula has a minus.

Worked example

Find sin 75° by writing it as sin(45° + 30°).

Use the angle-sum formula for sine.
$\sin (A + B) = \sin A \cos B + \cos A \sin B$
Substitute and work it out.
$\sin (A + B) = \sin (A) \cos (B) + \cos (A) \sin (B)$
$= \sin (45 °) \cos (30 °) + \cos (45 °) \sin (30 °)$
$= \sin (45 °) \cos (30 °) + \cos (45 °) \cdot 0.5$
$\approx 0.707107 \cdot 0.866025 + 0.707107 \cdot 0.5$
$\approx 0.965926$
The difference formula, sin(A − B) = sin A cos B − cos A sin B, gives sin(A − B) ≈ 0.258819.

Answer sin(45° + 30°) ≈ 0.9659

Why it works

One standard proof rotates a point on the unit circle by A and then by B and compares coordinates; another uses the distance between two points on the unit circle together with the law of cosines.

Common mistakes

Distributing the sine: $sin (A + B) ≠ sin A + sin B$ . At 45° and 30° that gives 1.207, more than any sine can be.
Getting the cosine sign wrong: $cos (A + B)$ has a minus between the terms, $cos (A − B)$ a plus.
Swapping sin and cos in the cross terms of the sine formula.

Angle sum and difference formulas

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes