Double Angle Formulae

students, trigonometry becomes much more powerful when you can rewrite one angle in terms of another. 📐 In this lesson, you will learn the double angle formulae, which let you express trigonometric values such as $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$ in terms of $\sin\theta$ and $\cos\theta$. These formulae are very useful in solving equations, simplifying expressions, and linking trigonometry to geometry and periodic motion.

Lesson objectives

By the end of this lesson, students, you should be able to:

explain the meaning of the double angle formulae and the notation involved,
use the formulae correctly in algebraic and trigonometric problems,
connect double angle ideas to geometry, graphs, and circular measure,
recognize when a trigonometric expression can be rewritten using a double angle identity,
solve IB-style questions that use these identities in reasoning and calculation.

A key idea in IB Mathematics Analysis and Approaches SL is that trigonometric identities are not just memorized facts. They are tools for transformation and problem solving. The double angle formulae are a great example because they connect several parts of the course: algebra, functions, graphs, and geometry. 🌟

What are double angle formulae?

A double angle formula gives the value of a trigonometric function of $2\theta$ in terms of the trigonometric function values at $\theta$. The three main identities are:

$$\sin(2\theta)=2\sin\theta\cos\theta$$

$$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

$$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$$

These are called identities because they are true for all angles where both sides are defined. That means you can replace one side with the other in an expression without changing its value.

It is important to notice that the formula for $\tan(2\theta)$ has a restriction: it is undefined when $1-\tan^2\theta=0$. This happens when $\tan\theta=\pm 1$. Also, the formula itself only makes sense when $\tan\theta$ is defined, so it is not used where cosine is $0$.

Where do the formulae come from?

The double angle formulae are based on angle addition formulae. For example, because $2\theta=\theta+\theta$, we can use the addition formula for sine:

$$\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta$$

Setting $\alpha=\beta=\theta$ gives:

$$\sin(2\theta)=\sin\theta\cos\theta+\cos\theta\sin\theta=2\sin\theta\cos\theta$$

This shows where the formula comes from, rather than treating it like magic ✨.

For cosine, the addition formula is:

$$\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta$$

Setting $\alpha=\beta=\theta$ gives:

$$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

For tangent, using the tangent addition formula:

$$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$$

and setting $\alpha=\beta=\theta$ gives:

$$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$$

Understanding these derivations helps you remember the formulae and use them more confidently in unfamiliar questions.

Useful forms of the cosine double angle identity

The identity

$$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

can also be rewritten in two other common forms using $\sin^2\theta+\cos^2\theta=1$.

If you replace $\sin^2\theta$ with $1-\cos^2\theta$, you get:

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

If you replace $\cos^2\theta$ with $1-\sin^2\theta$, you get:

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

These alternate forms are very useful. For example, if a question contains only $\sin\theta$, then $\cos(2\theta)=1-2\sin^2\theta$ may be the best choice. If it contains only $\cos\theta$, then $\cos(2\theta)=2\cos^2\theta-1$ may be easier.

Example 1: evaluating a double angle exactly

Suppose $\sin\theta=\frac{3}{5}$ and $\theta$ is in quadrant I. Find $\sin(2\theta)$.

First, use $\sin^2\theta+\cos^2\theta=1$ to find $\cos\theta$:

$$\cos\theta=\sqrt{1-\sin^2\theta}=\sqrt{1-\left(\frac{3}{5}\right)^2}=\frac{4}{5}$$

Since $\theta$ is in quadrant I, $\cos\theta$ is positive.

Now use the double angle formula:

$$\sin(2\theta)=2\sin\theta\cos\theta$$

Substitute the values:

$$\sin(2\theta)=2\cdot\frac{3}{5}\cdot\frac{4}{5}=\frac{24}{25}$$

So the exact value is $\frac{24}{25}$.

This type of question appears often in IB because it checks algebra, exact values, and quadrant reasoning together.

Example 2: simplifying an expression

Simplify $\cos^2\theta-\sin^2\theta$.

Using the identity directly, we get:

$$\cos^2\theta-\sin^2\theta=\cos(2\theta)$$

This is a simple example, but it shows a major exam skill: spotting an identity hidden inside an expression. Instead of expanding or manipulating everything separately, you can often compress the expression into a more elegant form.

This also matters when working with graphs. The function $y=\cos(2\theta)$ has a shorter period than $y=\cos\theta$, so rewriting an expression can help you understand how often a wave repeats. 📈

Example 3: solving a trigonometric equation

Solve $2\sin\theta\cos\theta=\frac{1}{2}$ for $0\leq\theta<2\pi$.

Recognize the left-hand side as $\sin(2\theta)$:

$$\sin(2\theta)=\frac{1}{2}$$

Now solve for $2\theta$:

$$2\theta=\frac{\pi}{6},\ \frac{5\pi}{6},\ \frac{13\pi}{6},\ \frac{17\pi}{6}$$

because these are the values in the interval $0\leq 2\theta<4\pi$ that satisfy $\sin x=\frac{1}{2}$.

Divide by $2$:

$$\theta=\frac{\pi}{12},\ \frac{5\pi}{12},\ \frac{13\pi}{12},\ \frac{17\pi}{12}$$

These are the solutions in the required interval.

Notice how the double angle identity turned a product into a single sine equation. That is a very common IB strategy: transform the problem into something easier to solve.

Geometry connections

Double angle formulae are closely related to geometry because they come from angles and rotations. In circular measure, angles are measured in radians, and trigonometric functions describe coordinates on the unit circle.

For a point on the unit circle at angle $\theta$, the coordinates are $(\cos\theta,\sin\theta)$. The point at angle $2\theta$ has coordinates $(\cos(2\theta),\sin(2\theta))$. The double angle formulae tell you how to find that new point from the old one.

This is useful in coordinate geometry too. For example, when a line or curve involves $\sin\theta$ and $\cos\theta$, identities can convert the expression into a form involving only one trigonometric function. That can simplify the equation of a curve or help with solving intersections.

The formulae also support reasoning about symmetry. Since $\sin(2\theta)=2\sin\theta\cos\theta$, its sign depends on the signs of both $\sin\theta$ and $\cos\theta$. That means it changes across quadrants in a way that matches the geometry of the unit circle.

Common mistakes to avoid

students, here are some common errors students make:

Writing $\cos(2\theta)=2\cos\theta-1$, which is incorrect. The correct form is $\cos(2\theta)=2\cos^2\theta-1$.
Forgetting to square the trig values in the cosine identity.
Using the tangent formula without checking whether it is defined.
Assuming $\sin(2\theta)=2\sin^2\theta\cos^2\theta$, which is not correct.
Losing exact values by converting to decimals too early.

A good habit is to ask: “Which identity matches the structure of the expression in front of me?” This helps you choose the right formula instead of guessing.

Conclusion

Double angle formulae are essential tools in IB Mathematics Analysis and Approaches SL. They allow you to rewrite trig expressions, solve equations, simplify identities, and connect algebra to geometry. The three key identities are

$$\sin(2\theta)=2\sin\theta\cos\theta$$

$$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

$$\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$$

and the cosine identity can also be written as

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

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

Knowing when and how to use each form is a major step toward strong trigonometric reasoning. With practice, these identities become a reliable shortcut for problem solving and a bridge to deeper geometric understanding. ✅

Study Notes

Double angle formulae express trigonometric values of $2\theta$ in terms of values at $\theta$.
The main identities are $\sin(2\theta)=2\sin\theta\cos\theta$, $\cos(2\theta)=\cos^2\theta-\sin^2\theta$, and $\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}$.
The cosine identity also has the useful forms $\cos(2\theta)=2\cos^2\theta-1$ and $\cos(2\theta)=1-2\sin^2\theta$.
These identities come from the angle addition formulae by setting both angles equal to $\theta$.
They are useful for exact values, simplifying expressions, solving equations, and graph analysis.
In problems, check the quadrant, the domain, and whether the expression matches a known identity.
Double angle formulae connect trigonometry with the unit circle, coordinate geometry, and circular measure.