45-45-90 Right Triangles• Consider the following 45-45-90 right triangle.

• Since the triangle is isosceles, the sides opposite the 45° angles are equal in measure. Assign a value of 1 to each side.

• Use the Pythagorean Theorem to determine the length of the hypotenuse.

2 2 2 a b c

2 2 21 1 c

22 c

2 c

• This 45-45-90 right triangle can give us the trigonometric function values of 45°.

oppsin45

hyp

adjcos45

hyp

opptan45

adj

1

2

1

2

11

1

• This leads us to some important values on the unit circle.

• Recall that on the unit circle we have …

( , )a b (cos ,sin )x x

• Consider the point (a,b) on the 45° ray of a unit circle.

• Since (a,b) = (cos 45°, sin 45°) , we have

• In radian form it would be …

1cos

4 2

1sin

4 2

• Moving around the unit circle with reference angles of π/4 we have …

Example 1:Find cos 3π/4

3 1cos

4 2

• Since cos x is equal to the first coordinate of the point we have …

Example 2:Find sin 5π/4

5 1sin

4 2

• Since sin x is equal to the second coordinate of the point we have …

Example 3:Find tan (-3π/4)

13 2tan

142

• Since tan x is equal to b/a we have …

1