Sum and Difference Identities

You know the unit circle gives sin and cos for every standard angle. But what about an angle that is not a standard one, like $75°$ or $\pi/12$? If you can write the awkward angle as a sum or difference of two standard angles, the addition formulas convert that algebraic decomposition into an exact value. Beyond computation, they are the engine behind the double-angle formulas, the half-angle identities used in calculus integration, the rotation matrices in linear algebra, and the proof that sine and cosine satisfy the differential equation $y'' + y = 0$.

The four formulas

For any angles $A$ and $B$:

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

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

$\cos(A + B) = \cos A\cos B - \sin A\sin B$

$\cos(A - B) = \cos A\cos B + \sin A\sin B$

Notice the sign pattern: sine of a sum mixes sine and cosine with a plus, while cosine of a sum mixes sine-and-sine with a minus. The minus inside is the formula’s claim to fame, it is what makes $\cos(A+B) \neq \cos A + \cos B$.

Worked example: cos 15°

$15° = 45° - 30°$, both standard. Apply the cosine difference formula:

$\cos 15° = \cos 45° \cos 30° + \sin 45° \sin 30° = \dfrac{\sqrt{2}}{2} \cdot \dfrac{\sqrt{3}}{2} + \dfrac{\sqrt{2}}{2} \cdot \dfrac{1}{2} = \dfrac{\sqrt{6} + \sqrt{2}}{4}$

Exact, no calculator. The same trick handles $\sin 75°$, $\tan 75°$, and any angle expressible as a sum or difference of $30°$, $45°$, $60°$, $90°$.

The double-angle formulas

Set $A = B = \theta$ in the sum formulas:

$\sin(2\theta) = 2\sin\theta\cos\theta$

$\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

The cosine version has three equivalent forms thanks to the Pythagorean identity. Each is the right one in a different context. Rearranged, they become the half-angle formulas $\cos^2\theta = (1 + \cos 2\theta)/2$ and $\sin^2\theta = (1 - \cos 2\theta)/2$, indispensable when integrating $\sin^2 x$ or $\cos^2 x$ in calculus.

Tangent of a sum

Dividing the sine sum by the cosine sum and simplifying yields:

$\tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$

valid whenever the denominator is nonzero. The denominator vanishes precisely when $A + B = \pi/2 + n\pi$, right where $\tan(A+B)$ has an asymptote, as it should.

• Signal Processing: Modulating a carrier signal with audio uses $\sin(\omega_c t + m(t))$