The Compound Angle Formulas

The sine of a sum is not the sum of the sines

It is tempting to think that sin(A + B) = sin A + sin B, as if the sine could be shared out over the two angles like a multiplication. Test the guess with A = 45° and B = 45°.

The left side is sin(45° + 45°) = sin 90° = 1. The right side is $\sin 45^{\circ }+\sin 45^{\circ }=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}=\frac{2}{\sqrt{2}}=\sqrt{2}$sin 45° + sin 45° = 1/√2 + 1/√2 = 2/√2 = √2, which is about 1.41. The two sides disagree, so the guess is wrong. In fact the right side cannot be the sine of anything: a sine is never more than 1.

It fails in radians as well. With A = 0.3 and B = 1.1, sin(0.3 + 1.1) = sin 1.4 = 0.9854, to 4 decimal places, but sin 0.3 + sin 1.1 = 0.2955 + 0.8912 = 1.1867.

With B = 45°: the gold curve is y = sin(x + 45°), and the plain curve is the guess, y = sin x + sin 45°, over one turn with one square across for every 90°. The plain straight line is y = 1. The gold curve never rises above it, as no sine can; the guess climbs to $1+\frac{1}{\sqrt{2}}$1 + 1/√2, about 1.71, at 90°.

The formula for sin(A + B)

The correct expansion is sin(A + B) = sin A cos B + cos A sin B. Each sine is paired with the cosine of the other angle, and the two products are added.

The formula is called a compound angle formula, because A + B is a compound angle, an angle made of two. It holds for every pair of angles A and B, in degrees or in radians.

Where the formula comes from

Draw a line of length 1 from the origin at the angle A + B above the horizontal. Its end is at height sin(A + B). Now draw a second line from the origin at the angle A, the first part of the compound angle, and drop a perpendicular onto it from the end of the first line. This makes a right triangle whose hypotenuse is 1 and whose angle at the origin is B. Its leg along the line at angle A has length cos B, and the perpendicular has length sin B.

Split the height sin(A + B) into two parts. The foot of the perpendicular is cos B along a line at angle A, so it stands at height cos B × sin A. The perpendicular itself is tilted: it makes the angle A with the vertical, so it rises by sin B × cos A. Adding the two parts, sin(A + B) = sin A cos B + cos A sin B.

The distance across gives the cosine formula in the same way. The foot is cos B × cos A across from the origin, and the perpendicular leans back to the left by sin B × sin A. So cos(A + B) = cos A cos B − sin A sin B, and the minus sign comes from that lean to the left.

A = 30° and B = 20°. The gold triangle has its angle A at the origin, and the green triangle, with angle B, is stacked on its hypotenuse. The height of the far corner splits into the gold part, sin A cos B = 0.47, and the green part, cos A sin B = 0.3, and together they make sin 50° = 0.77. Drag A and B: the two parts always add to the height. Raise both to 45° and the angle A + B is 90°, where the height is 1.

Checking the sine formula

Take A = 30° and B = 60°, so that A + B = 90° and the answer should be sin 90° = 1. The formula gives $\sin 30^{\circ }\cos 60^{\circ }+\cos 30^{\circ }\sin 60^{\circ }=\frac{1}{2}\times \frac{1}{2}+\left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right)=\frac{1}{4}+\frac{3}{4}=1$sin 30° cos 60° + cos 30° sin 60° = ½ × ½ + (√3/2)(√3/2) = 1/4 + 3/4 = 1, as it should.

At A = B = 45°, it gives $\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)+\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{\sqrt{2}}\right)=\frac{1}{2}+\frac{1}{2}=1$(1/√2)(1/√2) + (1/√2)(1/√2) = ½ + ½ = 1, which is sin 90°. The guess gave $\sqrt{2}$√2 for the same angles.

In radians, with A = 0.3 and B = 1.1: sin 0.3 cos 1.1 + cos 0.3 sin 1.1 = 0.1340 + 0.8514 = 0.9854, which is sin 1.4.

The formula for cos(A + B)

The cosine formula is cos(A + B) = cos A cos B − sin A sin B. Here cosine is paired with cosine and sine with sine, and the sign in the middle is a minus. That minus sign is the difference between the two formulas, and it is the one most often lost.

Check it at A = 30° and B = 60°: $\cos 30^{\circ }\cos 60^{\circ }-\sin 30^{\circ }\sin 60^{\circ }=\left(\frac{\sqrt{3}}{2}\right)\left(\frac{1}{2}\right)-\left(\frac{1}{2}\right)\left(\frac{\sqrt{3}}{2}\right)=0$cos 30° cos 60° − sin 30° sin 60° = (√3/2)(½) − (½)(√3/2) = 0, and cos 90° = 0. At A = B = 45° it gives ½ − ½ = 0 again.

In radians, cos 0.3 cos 1.1 − sin 0.3 sin 1.1 = 0.4333 − 0.2634 = 0.1700, which is cos 1.4. With a plus sign the result would be 0.6967, which is not cos 1.4.

Differences, and exact values

For a difference, replace B with −B. On the unit circle, the angle −B is the reflection of B in the x-axis, so cos(−B) = cos B and sin(−B) = −sin B. The middle signs flip: sin(A − B) = sin A cos B − cos A sin B, and cos(A − B) = cos A cos B + sin A sin B.

These give exact values for new angles. 15° is 45° − 30°, so $\sin 15^{\circ }=\sin 45^{\circ }\cos 30^{\circ }-\cos 45^{\circ }\sin 30^{\circ }=\left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right)-\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right)=\frac{\left(\sqrt{3}-1\right)}{\left(2\sqrt{2}\right)}$sin 15° = sin 45° cos 30° − cos 45° sin 30° = (1/√2)(√3/2) − (1/√2)(½) = (√3 − 1)/(2√2). Multiplying top and bottom by $\sqrt{2}$√2 gives $\frac{\left(\sqrt{6}-\sqrt{2}\right)}{4}$(√6 − √2)/4, which is 0.2588 to 4 decimal places, and a calculator gives sin 15° = 0.2588. In the same way $\cos 15^{\circ }=\frac{\left(\sqrt{6}+\sqrt{2}\right)}{4}=0.9659$cos 15° = (√6 + √2)/4 = 0.9659.

Dividing sin(A + B) by cos(A + B), then dividing top and bottom by cos A cos B, gives $\tan \left(A+B\right)=\frac{\left(\tan A+\tan B\right)}{\left(1-\tan A\tan B\right)}$tan(A + B) = (tan A + tan B)/(1 − tan A tan B). With A = 45° and B = 30°, $\tan 75^{\circ }=\frac{\left(1+\frac{1}{\sqrt{3}}\right)}{\left(1-\frac{1}{\sqrt{3}}\right)}=\frac{\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)}$tan 75° = (1 + 1/√3)/(1 − 1/√3) = (√3 + 1)/(√3 − 1), which simplifies to $2+\sqrt{3}$2 + √3, about 3.7321.

The usual mistakes

Splitting the sine: sin(A + B) = sin A + sin B. At 45° + 45° that gives $\sqrt{2}$√2, more than 1, but sin 90° is 1. No trigonometric function can be shared out over a sum.

Putting a plus sign in the cosine formula. cos(A + B) = cos A cos B − sin A sin B; only the sine formula adds its two products.

Pairing the wrong functions. The sine formula pairs each sine with the other angle's cosine; the cosine formula pairs cosine with cosine and sine with sine.

Stopping after one product. sin(30° + 60°) is ¼ + ¾ = 1; the ¾ on its own is only the cos 30° sin 60° part.

A car park barrier

In the application below, the arm of a barrier is raised to 75°. Without a calculator, 75° = 45° + 30° gives the height of its tip from the sine formula and the distance across from the cosine formula, with its minus sign.