Special Triangles

45°-45°-90° Triangles

If we're handling a 45° angle in trig, we need to be able to draw this reference triangle.

The trick? First, draw an isosceles right triangle. Isosceles triangles have two legs that are the same length.

Label each leg 1. Like this:

Then, thanks to the Pythagorean Theorem, we can find the hypotenuse:

1²+ 1² = c²
1 + 1 = c²
c² = 2

And since we've got one 45° angle and one 90° angle, we know the third angle must be 180° – 90° – 45° = 45°.

We clearly know a heck of a lot more than when we started. Not only do we know about trig functions, but we also know about their complementary functions. Go us. Now, here's how we apply it in a triangulated world.

When going from the hypotenuse to a leg → divide by .

When going from a leg to the hypotenuse → multiply by .

These tricks work for any 45°-45°-90° triangle, no matter how long its sides are. A triangle with two 45° angles and one 90° is always gonna be isosceles, which means it'll always have two equal legs.

Sample Problem

Find x.

First, rotate the reference triangle to match the problem triangle.

We have the leg. Now we need the hypotenuse. Multiply by . The answer is . Note that we also could've used the Pythagorean Theorem if we wanted, but this way was quicker.

Sample Problem

Find x.

Sketch your reference triangle next to your problem.

We have the hypotenuse. Now we need a leg. Divide by .

Now we just need to rationalize the denominator. You know how math teachers hate having a radical sign down in the basement.

Sample Problem

Find x.

This one is a cinch. Since we've got a 45° angle and a 90° angle, that must mean we have an isosceles triangle (equal-length legs, remember?). The answer is 5.

30°-60°-90° Triangles

If we're working with a 30° or 60° angle, we need to be able to draw this reference triangle.

To sketch this bad boy, first draw an equilateral triangle and label each side with a 2. Like this:

Now, split the triangle in half by bisecting the vertex angle. Label each half of the base with a 1. Like this:

Now, find the height, h. That's the red line. It's also the leg of a right triangle, so we can use our main dude Pythagoras.

1² + h² = 2²
h² = 2² – 1²
h²= 4 – 1
h²= 3

Victory. You've got your 30°-60°-90° reference triangle.

Here's how all this applies in the trig world.

The short leg is the length of the hypotenuse. Or, if you're an optimist, the hypotenuse is twice the length of the short leg. BTW, the short leg is always the side opposite the 30° angle.