Sums and products of sines and cosines
Learn how to rewrite sums and products of sines and cosines for Year 11 Specialist Mathematics in Queensland (QCAA). The product-to-sum identities turn a product of two ratios into a sum, and the sum-to-product identities turn a sum of two ratios into a product.
You will use these identities to simplify trigonometric expressions, find exact values, factorise to solve equations, and model beats — core skills in the trigonometric identities strand of the course.
Theory
Sums and products of sines and cosines are linked by two families of identities in Year 11 Specialist Mathematics (QCAA, Queensland). The product-to-sum identities turn a product such as \(2\sin A\cos B\) into a sum of sines or cosines, and the sum-to-product identities run the other way. This page shows both, with full worked examples for simplifying, finding exact values and solving equations.
Two waves added together can be rewritten as a single product, and a product of two trigonometric ratios can be rewritten as a sum. These conversions come straight from the angle-sum formulas.
The product-to-sum identities expand a product into a sum or difference: for example \(2\sin A\cos B=\sin(A+B)+\sin(A-B)\). Adding and subtracting the expansions of \(\sin(A\pm B)\) and \(\cos(A\pm B)\) produces all four.
The sum-to-product identities reverse this: a sum such as \(\sin P+\sin Q\) becomes \(2\sin\frac{P+Q}{2}\cos\frac{P-Q}{2}\). The new angles are the average \(\frac{P+Q}{2}\) and the half-difference \(\frac{P-Q}{2}\) of the original two.
These identities are the tools for simplifying trigonometric expressions, finding exact values like \(\cos 75^\circ+\cos 15^\circ\), factorising to solve equations, and modelling beats when two nearby frequencies combine.
Product-to-sum (multiply the angles out):
Sum-to-product (average \(S=\dfrac{P+Q}{2}\), half-difference \(D=\dfrac{P-Q}{2}\)):
Converting between sums and products
- Identify the form: a single product (two ratios multiplied) uses a product-to-sum identity; a sum or difference of two ratios uses a sum-to-product identity.
- Choose the matching identity by the ratio types (\(\sin\times\cos\), \(\cos\times\cos\), \(\sin\times\sin\)) and the sign between the terms.
- Substitute the angles: for product-to-sum use \(A+B\) and \(A-B\); for sum-to-product use the average \(\frac{P+Q}{2}\) and half-difference \(\frac{P-Q}{2}\).
- Simplify: evaluate any exact ratios, or factorise and set each factor to zero when solving an equation.
Match the identity \(2\sin A\cos B=\sin(A+B)+\sin(A-B)\) with \(A=4\theta\), \(B=2\theta\):
| \(2\sin 4\theta\cos 2\theta\) | \(=\) | \(\sin(4\theta+2\theta)+\sin(4\theta-2\theta)\) |
| \(=\) | \(\sin 6\theta+\sin 2\theta\) |
\(2\sin 4\theta\cos 2\theta=\sin 6\theta+\sin 2\theta\).
Use \(\cos P+\cos Q=2\cos\dfrac{P+Q}{2}\cos\dfrac{P-Q}{2}\) with \(P=7x\), \(Q=x\):
| \(\cos 7x+\cos x\) | \(=\) | \(2\cos\dfrac{7x+x}{2}\cos\dfrac{7x-x}{2}\) |
| \(=\) | \(2\cos\dfrac{8x}{2}\cos\dfrac{6x}{2}\) | |
| \(=\) | \(2\cos 4x\cos 3x\) |
\(\cos 7x+\cos x=2\cos 4x\cos 3x\).
Apply \(\sin P+\sin Q=2\sin\dfrac{P+Q}{2}\cos\dfrac{P-Q}{2}\):
| \(\sin 105^\circ+\sin 15^\circ\) | \(=\) | \(2\sin\dfrac{105^\circ+15^\circ}{2}\cos\dfrac{105^\circ-15^\circ}{2}\) |
| \(=\) | \(2\sin 60^\circ\cos 45^\circ\) | |
| \(=\) | \(2\times\dfrac{\sqrt3}{2}\times\dfrac{\sqrt2}{2}\) | |
| \(=\) | \(\dfrac{\sqrt6}{2}\) |
\(\sin 105^\circ+\sin 15^\circ=\dfrac{\sqrt6}{2}\).
Factorise the left side with sum-to-product:
| \(\cos 3x+\cos x\) | \(=\) | \(2\cos\dfrac{3x+x}{2}\cos\dfrac{3x-x}{2}\) |
| \(=\) | \(2\cos 2x\cos x\) |
Set the product to zero and solve each factor on \(0\le x<2\pi\):
| \(2\cos 2x\cos x\) | \(=\) | \(0\) |
| \(\cos 2x=0 \Rightarrow x\) | \(=\) | \(\dfrac{\pi}{4},\ \dfrac{3\pi}{4},\ \dfrac{5\pi}{4},\ \dfrac{7\pi}{4}\) |
| \(\cos x=0 \Rightarrow x\) | \(=\) | \(\dfrac{\pi}{2},\ \dfrac{3\pi}{2}\) |
\(x=\dfrac{\pi}{4},\ \dfrac{\pi}{2},\ \dfrac{3\pi}{4},\ \dfrac{5\pi}{4},\ \dfrac{3\pi}{2},\ \dfrac{7\pi}{4}\) (six solutions).
Common pitfalls
Frequently asked questions
What is the difference between product-to-sum and sum-to-product?
Product-to-sum turns a product like \(2\sin A\cos B\) into a sum or difference of sines or cosines. Sum-to-product does the reverse, turning \(\sin P+\sin Q\) into a product \(2\sin\frac{P+Q}{2}\cos\frac{P-Q}{2}\).
How do I remember the sum-to-product angles?
The two new angles are the average \(\frac{P+Q}{2}\) and the half-difference \(\frac{P-Q}{2}\) of the original angles \(P\) and \(Q\).
Where do these identities come from?
Add or subtract the angle-sum formulas. For example \(\sin(A+B)+\sin(A-B)=2\sin A\cos B\), because the \(\cos A\sin B\) terms cancel.
How do these identities help solve equations?
A sum such as \(\cos 3x+\cos x\) factorises into a product \(2\cos 2x\cos x\); setting each factor to zero gives all the solutions.
What are beats?
When two nearby frequencies add, \(\sin x+\sin 3x=2\sin 2x\cos x\) shows a fast carrier inside a slow envelope \(2\cos x\); the slow envelope is heard as a throbbing beat.