Double-Angle Formulas

Lesson notes

Overview

Doubling an angle is equivalent to a product of sine and cosine of the original. This topic is rated beginner and is number 48 of 245 in Mathematics. Core ideas and first intuitions. No prior knowledge needed.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

The double-angle formulas express the sine or cosine of 2θ in terms of θ: sin 2θ = 2 sin θ cos θ and cos 2θ = cos²θ − sin²θ. They come from the angle-addition formulas by setting both angles equal. Since cos 2θ has three equivalent forms, one can pick whichever is convenient. They help to simplify expressions, solve trigonometric equations, and explain why the range of a projectile is proportional to sin 2θ, which is largest at 45°.

Reading the relationship

The expression "sin 2θ = 2 sin θ cos θ" is a compact summary of the idea. It uses the symbols θ; name what each one stands for before you start. Notice that it contains a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Start by only moving one slider at a time. Say aloud what you expect, then check.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Pythagorean Identity”, the topic before it in Mathematics. Once you are comfortable, continue with “Inverse Trigonometric Functions”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.