This note covers the compound angle identities for sine, cosine and tangent, and the double angle identities that follow from them. They let you find exact values for angles such as 75°, simplify expressions and solve trigonometric equations that mix \(\theta\) and \(2\theta\).
Key ideas
It is tempting to think that \(\sin(A+B)\) equals \(\sin A+\sin B\). It does not. Try \(A=B=30^\circ\): \(\sin60^\circ\approx0.866\) but \(\sin30^\circ+\sin30^\circ=1\). The correct identities mix sines and cosines of the two angles.
The compound angle identities are:
- \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\)
- \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\). Note the sign change: plus on the left gives minus on the right.
- \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
Setting \(B=A\) in the "plus" versions gives the double angle identities. For sine, \(\sin2\theta=2\sin\theta\cos\theta\). For cosine there are three equivalent forms, because \(\sin^2\theta+\cos^2\theta=1\) lets you swap one square for the other:
- \(\cos2\theta=\cos^2\theta-\sin^2\theta\)
- \(\cos2\theta=2\cos^2\theta-1\)
- \(\cos2\theta=1-2\sin^2\theta\)
For tangent, \(\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}\).
Which form of \(\cos2\theta\) should you use? Choose the one that leaves only the function already in the problem. If an equation has \(\cos2x\) and \(\sin x\), use \(1-2\sin^2x\). If it has \(\cos2x\) and \(\cos x\), use \(2\cos^2x-1\). If it has \(\cos2x\) and \(\sin x\cos x\) or \(\sin2x\), the first form is often the best.
The identities also work in reverse. Seeing \(2\sin\theta\cos\theta\) in an expression, you can write \(\sin2\theta\). Seeing \(\cos^2\theta-\sin^2\theta\), you can write \(\cos2\theta\). The double angle idea also applies to any angle that is twice another, for example \(\sin4x=2\sin2x\cos2x\) and \(\cos x=1-2\sin^2\tfrac{x}{2}\).
To use these identities for exact values, split the angle into two angles whose values you know exactly, such as 30°, 45°, 60° or 90°. For example, \(75^\circ=45^\circ+30^\circ\) and \(15^\circ=45^\circ-30^\circ\).
When you are given one ratio and asked for others, draw a right-angled triangle or use \(\sin^2\theta+\cos^2\theta=1\). Then decide the sign of the missing ratio from the quadrant.
Rearranging the second and third forms of \(\cos2\theta\) gives two results that are very useful later in calculus: \(\cos^2\theta=\dfrac{1+\cos2\theta}{2}\) and \(\sin^2\theta=\dfrac{1-\cos2\theta}{2}\). They turn a squared function into a function of a double angle.
To see where the tangent identity comes from, divide the top and bottom of \(\dfrac{\sin(A+B)}{\cos(A+B)}\) by \(\cos A\cos B\). Each term then becomes a tangent. This also shows that the tangent identities are undefined if either \(\cos A\), \(\cos B\) or \(\cos(A\pm B)\) is zero, or if the denominator is zero.
Strategy for proving an identity: start with the more complicated side and work towards the other side. Do not move terms across the equals sign. Look for a double angle that can be expanded, a product that can be recognised as a double angle, or a \(1\) that can be replaced using \(\sin^2\theta+\cos^2\theta=1\). If you are stuck, write everything in terms of sine and cosine.
Strategy for solving an equation: decide whether the angles are all the same. If you see both \(x\) and \(2x\), use a double angle identity to get a single angle. Then look for a quadratic in one trigonometric function, or a common factor. Never divide both sides by a trigonometric expression that could be zero, because you will lose solutions. For example, in \(\sin2x=\sin x\), write \(2\sin x\cos x-\sin x=0\) and factorise as \(\sin x(2\cos x-1)=0\).
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B\) | Sine of a sum or difference; exact values; combining waves. |
| \(\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B\) | Cosine of a sum or difference; the sign flips. |
| \(\tan(A\pm B)=\dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B}\) | Tangent of a sum or difference; useful when tangents are given. |
| \(\sin2\theta=2\sin\theta\cos\theta\) | Double angle for sine. |
| \(\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta\) | Double angle for cosine; pick the form that suits the problem. |
| \(\tan2\theta=\dfrac{2\tan\theta}{1-\tan^2\theta}\) | Double angle for tangent. |
Worked examples
Example 1 Find the exact value of \(\sin75^\circ\).
Write \(75^\circ=45^\circ+30^\circ\).
\(\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ\)
\(=\dfrac{\sqrt2}{2}\cdot\dfrac{\sqrt3}{2}+\dfrac{\sqrt2}{2}\cdot\dfrac12=\dfrac{\sqrt6}{4}+\dfrac{\sqrt2}{4}\)
Check: \(\dfrac{2.449+1.414}{4}=0.966\), which matches \(\sin75^\circ\) on a calculator.
Answer: \(\sin75^\circ=\dfrac{\sqrt6+\sqrt2}{4}\).
Example 2 The angle \(\theta\) is obtuse and \(\cos\theta=-\tfrac45\). Find \(\sin2\theta\) and \(\cos2\theta\).
First find \(\sin\theta\). From \(\sin^2\theta=1-\cos^2\theta=1-\tfrac{16}{25}=\tfrac{9}{25}\), \(\sin\theta=\pm\tfrac35\). An obtuse angle has positive sine, so \(\sin\theta=\tfrac35\).
\(\sin2\theta=2\sin\theta\cos\theta=2\times\tfrac35\times\left(-\tfrac45\right)=-\tfrac{24}{25}\).
\(\cos2\theta=2\cos^2\theta-1=2\times\tfrac{16}{25}-1=\tfrac{32}{25}-1=\tfrac{7}{25}\).
Check: \(\sin^22\theta+\cos^22\theta=\tfrac{576}{625}+\tfrac{49}{625}=1\).
Answer: \(\sin2\theta=-\dfrac{24}{25}\), \(\cos2\theta=\dfrac{7}{25}\).
Example 3 Solve \(\cos2x=3\sin x\) for \(0\le x\le2\pi\). Give answers in radians to 3 s.f.
The equation contains \(\sin x\), so use \(\cos2x=1-2\sin^2x\).
\(1-2\sin^2x=3\sin x\), so \(2\sin^2x+3\sin x-1=0\).
This is a quadratic in \(\sin x\). Use the quadratic formula (or the polynomial solver on your GDC): \(\sin x=\dfrac{-3\pm\sqrt{9+8}}{4}=\dfrac{-3\pm\sqrt{17}}{4}\).
The two values are \(0.2808\) and \(-1.781\). The second is impossible because \(\sin x\ge-1\).
For \(\sin x=0.2808\): \(x=\arcsin0.2808=0.2846\). Sine is also positive in the second quadrant, so \(x=\pi-0.2846=2.857\).
Answer: \(x=0.285\) or \(x=2.86\) (3 s.f.).
Common errors
- Writing \(\sin(A+B)=\sin A+\sin B\), or \(\cos2\theta=2\cos\theta\). Neither is true.
- Getting the sign wrong in \(\cos(A\pm B)\). The sign between the two terms is the opposite of the sign inside the bracket.
- Forgetting to reject impossible values, such as \(\sin x=-1.78\), after solving a quadratic.
- Choosing the wrong form of \(\cos2\theta\) and ending up with both \(\sin x\) and \(\cos x\) in the same equation.
- Finding only the calculator value of an inverse function and missing the second solution in the interval.
Exam tip: When a question says "show that" or asks for an exact value, write the identity you use first and substitute the exact values line by line. Do not jump to a decimal.
Check yourself
- Find the exact value of \(\cos15^\circ\).
Answer
\(\cos(45^\circ-30^\circ)=\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ=\tfrac{\sqrt6}{4}+\tfrac{\sqrt2}{4}=\tfrac{\sqrt6+\sqrt2}{4}\). - Given \(\tan A=2\) and \(\tan B=\tfrac13\) with both angles acute, find \(\tan(A+B)\).
Answer
\(\tan(A+B)=\dfrac{2+\frac13}{1-2\times\frac13}=\dfrac{\frac73}{\frac13}=7\). - Given \(\sin x=\tfrac{1}{3}\), find \(\cos2x\).
Answer
\(\cos2x=1-2\sin^2x=1-2\times\tfrac19=\tfrac79\). - Simplify \(\dfrac{\sin2x}{1+\cos2x}\).
Answer
The numerator is \(2\sin x\cos x\). Using \(1+\cos2x=2\cos^2x\), the denominator is \(2\cos^2x\). The fraction is \(\dfrac{\sin x}{\cos x}=\tan x\).