Prove sin^2 x + cos^2 x = 1

Step by step

The question

Question: show sin² x + cos² x = 1.

This is true for every angle x.

1 Draw the unit circle

The unit circle has radius 1.

The point P at angle x is (cos x, sin x).

Draw a line from P down to the x-axis.

Across is cos x.

Up is sin x.

The longest side is the radius: 1.

Careful!

sin squared x is the sine of x, then squared.

Not the sine of x squared.

2 Use Pythagoras

It is a right triangle.

Pythagoras: across squared plus up squared is the longest side squared.

Books write (cos x)² as cos² x.

And 1 squared is 1.

Swap the order of adding: sin² x + cos² x = 1.

True for every angle!

3 Check

Let's check with a number: x = 30°.

sin 30° is 0.5.

cos 30° is about 0.8660.

Squared: 0.25 and 0.75.

Together: 1.

Let's look back.

First: draw the unit circle.

Then: use Pythagoras.

Last: check.