What Are the Double Angle Identities for Sine and Cosine?

The double angle identities are $\sin(2\theta)=2\sin\theta\cos\theta$ and $\cos(2\theta)=\cos^2\theta-\sin^2\theta$. The cosine identity also has two equivalent forms: $\cos(2\theta)=2\cos^2\theta-1=1-2\sin^2\theta$. They convert trigonometric expressions involving $2\theta$ into expressions involving $\theta$, or reverse that process.

These identities are IB Mathematics: Analysis and Approaches SL 3.6 core content for both AA SL and AA HL. They may appear on Paper 1 or Paper 2; Paper 1 requires exact, non-calculator working, while Paper 2 may also use technology.

At HL, these results can also be derived from the compound-angle identities. Set $\alpha=\theta$ and $\beta=\theta$ in $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$ to obtain the sine result. Likewise, substituting into gives the first cosine form. At SL, you should know and apply the identities directly; the derivation is a useful connection rather than a separate requirement. This explains why every term uses the same angle and why squares appear in cosine.

The Identities

The three cosine forms follow from the Pythagorean identity, $\sin^2\theta+\cos^2\theta=1$. Starting with $\cos(2\theta)=\cos^2\theta-\sin^2\theta$, replace by :

$$\cos(2\theta)=\cos^2\theta-(1-\cos^2\theta)=2\cos^2\theta-1.$$

Alternatively, replacing $\cos^2\theta$ by $1-\sin^2\theta$ gives $\cos(2\theta)=1-2\sin^2\theta$. Therefore, choose the version that matches the information supplied and minimizes unnecessary algebra.

Worked Example

Suppose $\sin\theta=\frac{3}{5}$, and $\theta$ is acute. From $\sin^2\theta+\cos^2\theta=1$,

$$\cos\theta=\frac{4}{5}.$$

Then use both identities:

$$\sin(2\theta)=2\left(\frac{3}{5}\right)\left(\frac{4}{5}\right)=\frac{24}{25},$$ $$\cos(2\theta)=1-2\left(\frac{3}{5}\right)^2=\frac{7}{25}.$$

The acute condition makes $\cos\theta$ positive; without quadrant information, its sign might be negative, changing $\sin(2\theta)$ but not the cosine-only calculation.

A common misconception is $\sin(2\theta)=2\sin\theta$. This is false because a trigonometric function is not linear: doubling its input does not generally double its output. Another error is dropping squares from $\cos^2\theta-\sin^2\theta$.

Exam Technique

In an IB show that or simplify question, state the identity before substituting and show the algebraic steps. Select the form containing the given ratio, check the quadrant before choosing a square-root sign, and give exact Paper 1 values unless instructed otherwise. Radians are assumed unless degrees are specified.