3.10 Compound angle identities

A compound angle is an angle written as a sum or difference of two others, such as $A + B$ or $\theta - \frac{\pi}{3}$. The compound angle identities give the sine, cosine and tangent of such an angle from the sines, cosines and tangents of the separate parts. They produce exact values for angles such as $15^\circ$

The same identities explain why two waves of the same frequency always add up to a single wave of that frequency. A combination such as $3\cos\theta + 4\sin\theta$ is one cosine wave with amplitude $5$, shifted sideways. This fact is used wherever oscillations combine, from sound and alternating current to tides.

Why sin(A + B) is not sin A + sin B

A function $f$ that satisfies $f(A + B) = f(A) + f(B)$ for all inputs is called linear, and sine is not one. Taking $A = B = \frac{\pi}{4}$

Since $1 \ne \sqrt{2}$, the sine of a sum is not the sum of the sines. The correct rule involves both sines and cosines of the two angles.

The compound angle identities

The six compound angle identities are usually written in three lines, with the upper sign of $\pm$ and $\mp$ going together and the lower signs going together.

Written out in full, the six identities are:

The sine identities keep the sign of the compound angle and mix the functions (sin cos, then cos sin). The cosine identities reverse the sign and keep the functions together (cos cos, then sin sin).

Deriving cos(A − B) with the distance formula

All six identities follow from one of them, $\cos(A - B) = \cos A\cos B + \sin A\sin B$, which can be proved on the unit circle. Take the points $P(\cos A, \sin A)$

First, find $PQ^2$ with the distance formula:

Now rotate the triangle about $O$ through the angle $-B$. The point $Q$ moves to $Q'(1, 0)$ and $P$

Setting the two expressions equal and cancelling gives

The argument uses only coordinates on the unit circle, so it holds for all angles $A$ and $B$, not only acute ones. The cosine rule in triangle $OPQ$ gives the same result for $PQ^2$ when the triangle exists.

Deriving the other identities

The remaining identities follow from the first by replacing angles and using the symmetry of the graphs. Cosine is an even function and sine is an odd function, so $\cos(-B) = \cos B$ and $\sin(-B) = -\sin B$.

1. Cosine of a sum. Replace $B$ with $-B$:

2. Sine of a sum. Use $\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right)$ and $\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$

3. Sine of a difference. Replace $B$ with $-B$ in step 2: $\sin(A - B) = \sin A\cos B - \cos A\sin B$

5. Tangent of a difference. Replace $B$ with $-B$, using $\tan(-B) = -\tan B$.

A second proof with complex numbers

Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ from 1.13 gives the sum identities in a few lines. By the laws of exponents, $e^{i(A + B)} = e^{iA}e^{iB}$

Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal. The real parts give $\cos(A + B)$ and the imaginary parts give $\sin(A + B)$. This is an example of equivalence between representations: the same fact about angles appears as a property of rotation in the plane and as a law of exponents.

Exact values from compound angles

Angles such as $15^\circ$, $75^\circ$ and $105^\circ$ are sums or differences of $30^\circ$

Some angles can be split in more than one way, for example $75^\circ = 45^\circ + 30^\circ$ or $120^\circ - 45^\circ$

Using given ratios

When the ratios of $A$ and $B$ are given instead of the angles themselves, find the missing sine or cosine of each angle first, with the sign fixed by its quadrant, then substitute.

Recognising an identity in reverse

An expression that has the shape of the right-hand side of an identity can be collapsed into a single function. This is often the quickest route to an exact value.

Double angle identities

Deriving the double angle identities

Setting $B = A$ in the sum identities gives the double angle identities. The ones for sine and cosine are studied in 3.6; here they appear as special cases.

The same substitution in the tangent identity gives a new result.

Triple angles and other multiples

Applying the identities twice gives the trigonometric functions of $3\theta$ and other multiples in terms of $\theta$. The method is to write the multiple as a sum, for example $3\theta = 2\theta + \theta$, and expand.

Proving identities with compound angles

To prove an identity that contains a compound angle, expand it with the appropriate identity first. Then simplify, aiming at the form on the other side. The target suggests the route: if it contains only $\tan$, divide by a product of cosines; if it contains only $\sin$ and $\cos$ of single angles, expand and collect like terms.

Results such as this one turn a sum of two sines into a product, which is useful in integration (5.16) and in physics, where two notes of slightly different frequencies produce beats.

When one of the angles is a standard angle, substitute its exact values as soon as the identity has been expanded.

Solving equations with compound angles

An equation that contains a compound angle with a known part, such as $\sin\left(x + \frac{\pi}{6}\right)$, is usually solved by expanding, collecting the $\sin x$ terms and the $\cos x$

The same method gives exact answers even when the tangent is not a familiar value, provided it matches a compound angle found earlier.

The tangent identity turns an equation in $\tan(x + \text{constant})$ into an equation in $\tan x$ alone, often a quadratic.

Writing a cos θ + b sin θ as a single wave

The harmonic form R cos(θ − α)

The graph of $y = 3\cos\theta + 4\sin\theta$ looks exactly like a cosine curve: it has the same period, $2\pi$, but a larger amplitude and a horizontal shift. The compound angle identity explains why. For $R > 0$,

This equals $a\cos\theta + b\sin\theta$ for all $\theta$ if the coefficients match:

Squaring and adding gives $R^2(\cos^2\alpha + \sin^2\alpha) = a^2 + b^2$

The harmonic form shows at once that $a\cos\theta + b\sin\theta$ has maximum value $R$, when $\theta = \alpha$, and minimum value $-R$

The values of $R\cos\alpha$ and $R\sin\alpha$ fix the quadrant of $\alpha$, so check their signs before using $\arctan$.

Combining waves

Any two oscillations of the same period combine into one oscillation of that period. If two sound waves of the same frequency reach a microphone, contributing pressures $p_1 = a\cos\omega t$ and $p_2 = b\sin\omega t$

Exam-style questions on compound angles

Self-check

• To do:

  I can state the compound angle identities for $\sin(A \pm B)$, $\cos(A \pm B)$ and $\tan(A \pm B)$