Pythagoras Theorem

For the next trigonometric identities we start with Pythagoras' Theorem:

The Pythagorean Theorem says that, in a right triangle, the square of a plus the square of b is equal to the square of c: a2 + b2 = c2

Dividing through by c² gives

a²c² + b²c² = c²c²

This can be simplified to:

(ac)² + (bc)² = 1

Now, a/c is Opposite / Hypotenuse, which is sin(θ)

And b/c is Adjacent / Hypotenuse, which is cos(θ)

So (a/c)² + (b/c)² = 1 can also be written:

sin² θ + cos² θ = 1

Note:

• sin² θ means to find the sine of θ, then square the result, and
• sin θ² means to square θ, then do the sine function

Example: 32°

Using 4 decimal places only:

• sin(32°) = 0.5299...
• cos(32°) = 0.8480...

Now let's calculate sin² θ + cos² θ:

0.5299² + 0.8480²
= 0.2808... + 0.7191...
= 0.9999...

We get very close to 1 using only 4 decimal places. Try it on your calculator, you might get better results!