30-60-90 Right Triangles

1. 30-60-90 Right Triangles • Consider the following equilateral triangle, with each side having a value of 2. • Drop a perpendicular segment from the top vertex to the base side. • The base side has been bisected into two segments of length 1.

2. Since the angle at the top of the triangle has been bisected, the original 60° angle has been split into new angles that are 30° each. • Since the original triangle was equilateral, the base angles are 60°each.

3. This is a 30-60-90 right triangle. • Consider the new triangle formed on the left side. • We have just proven that in a 30-60-90 triangle the short leg(always opposite the 30° angle) is always half the length of the hypotenuse.

4. Name the long leg b and determine its value.

5. This 30-60-90 right triangle can give us the trigonometric function values of 30°and 60°. We first do the 30° angle.

6. We now do the 60° angle.

7. Recall that on the unit circle we have … • This leads us to some important values on the unit circle.

8. Since (a,b) = (cos 30°, sin 30°) , we have • Consider the point (a,b) on the 30° ray of a unit circle.

9. In radian form it would be …

10. Moving around the unit circle with reference angles of π/6 we have …

11. Example 1: • Find cos 5π/6 • Since cosx is equal to the first coordinate of the point we have …

12. Example 2: • Find sin 7π/6 • Since sin x is equal to the second coordinate of the point we have …

13. Example 3: • Find tan (-7π/6) • Since tan x is equal to b/a we have …

14. In similar fashion, moving around the unit circle with reference angles of π/3 (60°)we have …

15. Example 1: • Find cos 4π/3 • Since cosx is equal to the first coordinate of the point we have …

16. Example 2: • Find sin -2π/3 • Since sin x is equal to the second coordinate of the point we have …

17. Example 3: • Find tan (2π/3) • Since tan x is equal to b/a we have …