IB Maths AA 3.9 Notes

Compound angle formulas

Compound angle identities let us find the trigonometric ratio of a sum or difference of two angles. They are useful for exact values, simplifying expressions, proving identities and solving equations.

The main formulas are:

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

$\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B$

$\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}$

Notice that the signs in the sine formula match, but in the cosine and tangent formulas they are opposite.

If $A=B=x$, then the tangent compound angle formula gives:

$\tan2x=\frac{2\tan x}{1-\tan^2x}$

This is the double angle identity for tangent.

The double angle identities for sine and cosine are also obtained by setting $A=B=x$:

$\sin2x=2\sin x\cos x$

$\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x$

Thus, to summarise:

• For sine: $\sin(2x)=\sin(x+x)=\sin x\cos x+\cos x\sin x=2\sin x\cos x$
• For cosine: $\cos(2x)=\cos(x+x)=\cos^2x-\sin^2x$
• For tangent: $\tan(2x)=\tan(x+x)=\frac{2\tan x}{1-\tan^2x}$