Angle sum and difference identities
Master the angle sum and difference identities for Year 11 Specialist Mathematics in Queensland (QCAA). These rules expand the sine, cosine and tangent of a sum or difference of two angles into products of their individual sines and cosines.
You will learn to find the exact values of non-standard angles such as fifteen and seventy-five degrees, evaluate and simplify trigonometric expressions, and prove new identities — the groundwork for the double-angle formulae and wave form later in the course.
Theory
The angle sum and difference identities expand the sine, cosine and tangent of \(A\pm B\) into products of the individual sines and cosines. In Year 11 Specialist Mathematics (QCAA, Queensland) they let you find exact values of non-standard angles such as \(15^\circ\) and \(75^\circ\), simplify expressions, and prove further identities.
The angle sum and difference identities rewrite a trigonometric function of a combined angle \(A\pm B\) using only the sines and cosines of \(A\) and \(B\) on their own. They are exact algebraic rules that hold for every value of \(A\) and \(B\).
For sine and cosine: \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\) and \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\). The sine expansion keeps the \(\pm\) sign; the cosine expansion flips it, so a plus outside becomes a minus inside.
For tangent: \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\). The denominator takes the opposite sign to the numerator.
Two everyday uses drive the topic. To find an exact value, split the angle into two special angles, e.g. \(75^\circ=45^\circ+30^\circ\). When a ratio is given, e.g. \(\sin A=\dfrac35\) with \(A\) acute, read the missing side from a right triangle (here \(\cos A=\dfrac45\)) before substituting.
The sine sum and difference (same sign throughout):
The cosine sum and difference (the sign flips inside):
The tangent sum and difference:
Finding an exact value with an identity
- Split the angle into two angles whose sine, cosine and tangent you already know, such as \(15^\circ=45^\circ-30^\circ\) or \(105^\circ=60^\circ+45^\circ\).
- Choose the identity that matches the function and the sign, and write it out with \(A\) and \(B\) as your two angles.
- Substitute the exact ratios for each special angle from the \(45^\circ\) or \(30\text{-}60\text{-}90\) triangle.
- Simplify to a single exact surd or fraction, rationalising the denominator for a tangent value.
Split \(15^\circ=45^\circ-30^\circ\) and use the sine difference identity, showing every stage:
| \(\sin 15^\circ\) | \(=\) | \(\sin(45^\circ-30^\circ)\) |
| \(=\) | \(\sin 45^\circ\cos 30^\circ-\cos 45^\circ\sin 30^\circ\) | |
| \(=\) | \(\dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2}-\dfrac{\sqrt2}{2}\cdot\dfrac12\) | |
| \(=\) | \(\dfrac{\sqrt6}{4}-\dfrac{\sqrt2}{4}\) | |
| \(=\) | \(\dfrac{\sqrt6-\sqrt2}{4}\) |
\(\sin 15^\circ=\dfrac{\sqrt6-\sqrt2}{4}\).
Split \(75^\circ=45^\circ+30^\circ\), apply the tangent sum identity, then rationalise:
| \(\tan 75^\circ\) | \(=\) | \(\tan(45^\circ+30^\circ)\) |
| \(=\) | \(\dfrac{\tan 45^\circ+\tan 30^\circ}{1-\tan 45^\circ\tan 30^\circ}\) | |
| \(=\) | \(\dfrac{1+\dfrac{1}{\sqrt3}}{1-\dfrac{1}{\sqrt3}}\) | |
| \(=\) | \(\dfrac{\sqrt3+1}{\sqrt3-1}\) | |
| \(=\) | \(\dfrac{(\sqrt3+1)^2}{(\sqrt3-1)(\sqrt3+1)}\) | |
| \(=\) | \(\dfrac{4+2\sqrt3}{2}\) | |
| \(=\) | \(2+\sqrt3\) |
\(\tan 75^\circ=2+\sqrt3\).
Read the sine sum identity right-to-left, then evaluate the special angle:
| \(\sin 25^\circ\cos 5^\circ+\cos 25^\circ\sin 5^\circ\) | \(=\) | \(\sin(25^\circ+5^\circ)\) |
| \(=\) | \(\sin 30^\circ\) | |
| \(=\) | \(\dfrac12\) |
The value is \(\dfrac12\).
First read the missing ratio for each angle from its right triangle:
| \(\sin A\) | \(=\) | \(\dfrac{15}{17}\) |
| \(\cos B\) | \(=\) | \(\dfrac45\) |
Now apply the cosine difference identity, one operation per line:
| \(\cos(A-B)\) | \(=\) | \(\cos A\cos B+\sin A\sin B\) |
| \(=\) | \(\dfrac{8}{17}\cdot\dfrac45+\dfrac{15}{17}\cdot\dfrac35\) | |
| \(=\) | \(\dfrac{32}{85}+\dfrac{45}{85}\) | |
| \(=\) | \(\dfrac{77}{85}\) |
\(\cos(A-B)=\dfrac{77}{85}\).
Common pitfalls
Frequently asked questions
What are the angle sum and difference identities?
They expand \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\), \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\) and \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\).
How do you find the exact value of cos 15 degrees?
Write \(15^\circ=45^\circ-30^\circ\) and use \(\cos(A-B)=\cos A\cos B+\sin A\sin B\), which gives \(\cos 15^\circ=\dfrac{\sqrt6+\sqrt2}{4}\).
Why does the cosine identity have a minus sign?
For \(\cos(A+B)\) the sign inside flips: the expansion is \(\cos A\cos B-\sin A\sin B\). For \(\cos(A-B)\) it becomes a plus.
Is sin(A + B) the same as sin A + sin B?
No. Sine does not distribute over addition. You must use \(\sin(A+B)=\sin A\cos B+\cos A\sin B\).
How do you use a given ratio like sin A = 3/5?
Sketch a right triangle: opposite \(3\), hypotenuse \(5\), so the adjacent side is \(4\) and \(\cos A=\dfrac45\). Fix the sign from the quadrant, then substitute into the identity.
When do you read an identity backwards?
When you see the pattern \(\sin A\cos B\pm\cos A\sin B\) or \(\cos A\cos B\mp\sin A\sin B\), collapse it to a single sine or cosine of the combined angle.