Pythagorean Identity

Lesson notes

Overview

Squares of sine and cosine always add to one — the circle equation in disguise. This topic is rated intermediate and is number 47 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

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 identity sin²θ + cos²θ = 1 is the equation of the unit circle written in trigonometry: the coordinates (cos θ, sin θ) must satisfy x² + y² = 1. It holds for every angle. If you know one of the two values and the quadrant, you can find the other. Dividing the identity through by cos²θ or sin²θ gives two companions, 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ, which are handy for simplifying integrals.

Reading the relationship

The expression "sin²θ + cos²θ = 1" 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 squared term, so doubling that quantity changes the result by a factor of four, and 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

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

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 “Amplitude, Period and Phase”, the topic before it in Mathematics. Once you are comfortable, continue with “Double-Angle Formulas”.

Why it matters

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