How to Calculate Compound Angles in Trigonometry

In trigonometry, a compound angle is an angle written as a sum or difference, such as $75^\circ=45^\circ+30^\circ$. To calculate its sine, cosine, or tangent, use the compound-angle identities and choose smaller angles whose trig values you know.

This article uses “compound angle” in that standard trigonometric sense. If you mean a geometric angle made from adjacent parts, add the parts when they share a vertex and are measured in the same direction; the identities below apply when you need a trigonometric value of the combined angle.

The compound-angle identities

For angles $A$ and $B$, the sum and difference formulas are:

$$\begin{aligned} \sin(A+B)&=\sin A\cos B+\cos A\sin B,\\ \sin(A-B)&=\sin A\cos B-\cos A\sin B,\\ \cos(A+B)&=\cos A\cos B-\sin A\sin B,\\ \cos(A-B)&=\cos A\cos B+\sin A\sin B. \end{aligned}$$

The tangent formulas are:

$$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B},\qquad \tan(A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}.$$

The tangent formulas apply when the expressions are defined; in particular, the denominator must not be zero. You can also calculate tangent from sine divided by cosine.

A useful way to remember the signs is that sine keeps the sign between its two terms: plus for a sum, minus for a difference. Cosine has a minus between the terms for a sum and a plus for a difference. The tangent formulas have different denominators, so check them rather than guessing.

A step-by-step method

1. Rewrite the angle as a sum or difference of familiar angles. Common choices include $30^\circ$, $45^\circ$, and $60^\circ$, or their radian equivalents $\pi/6$, $\pi/4$, and $\pi/3$.
2. Choose the formula for sine, cosine, or tangent. Use the sum formula if the decomposition uses $A+B$ and the difference formula if it uses $A-B$.
3. Substitute known values. Keep the factors in the same order as the formula, and include signs such as $\sin(-x)=-\sin x$ when needed.
4. Simplify the exact expression. If a decimal is requested, evaluate it only after simplifying.
5. Check the result. The sign should agree with the quadrant of the original angle, and the value of sine or cosine must lie between $-1$ and $1$.

You can use degrees or radians, but all angles in one calculation must use the same unit.

Worked example: calculate $\sin 75^\circ$

Choose $75^\circ=45^\circ+30^\circ$. The sine sum identity gives

$$\begin{aligned} \sin 75^\circ &=\sin(45^\circ+30^\circ)\\ &=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ. \end{aligned}$$

Use the exact values $\sin45^\circ=\cos45^\circ=\sqrt{2}/2$, $\cos30^\circ=\sqrt{3}/2$, and $\sin30^\circ=1/2$:

$$\begin{aligned} \sin75^\circ &=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\cdot\frac{1}{2}\\ &=\frac{\sqrt{6}+\sqrt{2}}{4}\\ &\approx 0.966. \end{aligned}$$

Since $75^\circ$ is in the first quadrant, its sine should be positive. The decimal is also less than $1$ and close to $1$, as expected for an angle near $90^\circ$.

The same decomposition gives cosine using the cosine sum identity:

$$\cos75^\circ=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ=\frac{\sqrt{6}-\sqrt{2}}{4}.$$

For tangent, divide these results or use the tangent formula. For example,

$$\tan75^\circ=\frac{\tan45^\circ+\tan30^\circ}{1-\tan45^\circ\tan30^\circ}=2+\sqrt{3}.$$

Common mistakes

• Using the wrong cosine sign. The middle sign in $\cos(A+B)$ is minus, while in $\cos(A-B)$ it is plus.
• Treating the identity as a formula for the angle itself. These identities calculate a trig function of a sum or difference; they do not say that $\sin(A+B)=\sin A+\sin B$.
• Mixing degrees and radians. Do not combine, for example, $45^\circ$ and $\pi/6$ without converting one of them first.
• Ignoring quadrant signs. Exact values for familiar reference angles may be positive or negative depending on the actual angle. For instance, $\sin150^\circ=\sin(180^\circ-30^\circ)=1/2$, while $\cos150^\circ=-\sqrt{3}/2$.
• Using tangent when its denominator is zero. If the tangent sum formula has a zero denominator, tangent of the combined angle is undefined; calculate sine and cosine separately if useful.