1. Special Right Triangles Definition and use

2. The 30-60-90 Triangle Definition  There are many right angle triangles. Today we are most interested in right angle triangles that have measures of either 30-60-90 or 45-45-90 for their angles.  We call these triangles special triangles because of their unique nature.  These triangles can be solved through their side ratios instead of resorting to the Pythagorean equation or Trig functions.

3. The 30-60-90 Triangle Definition  Let’s examine the 30-60-90 triangle first. The 30-60-90 angle measures create memorable side ratios. The sides of this triangle are termed the short leg, long leg and hypotenuse.

4. The 30-60-90 Triangle Examples  Let’s see some examples of using the leg ratios to find leg values.

5. The 30-60-90 Triangle Examples  Let’s see another example!

6. The 30-60-90 Triangle Examples

7. The 30-60-90 Triangle Examples  These ratios hold no matter what the orientation of the triangle, just like the relationships of “opposite” and “adjacent”.

8. The 45-45-90 Triangle Definition  Next let’s examine the 45-45-90 triangle. The 45-45-90 angle measures create equal length sides. The sides of this triangle are simply termed the legs and hypotenuse.

9. The 45-45-90 Triangle Examples  Next let’s see an example. If a leg measures 4, what is the other leg and hypotenuse?

10. The 45-45-90 Triangle Examples

11. The 45-45-90 Triangle Examples  If the hypotenuse measures 6, what the measures of the legs?

12. Finding the leg lengths in special triangles  In the triangle below, what are the measures of the short leg and long leg if the hypotenuse is 14?

13. Finding the leg lengths in special triangles

14. Identifying opposite and adjacent sides

16. Trigonometry  If the measure of angle A is 40°and the hypotenuse (side c) is 17, what are the values of a and b?  What are the trig function definitions?  SOH  CAH  TOA Trigonometric ratios

17. Trigonometry  If the measure of angle B is 12°and side a is 4, what are the values of b and c?  What are the trig function definitions?  SOH  CAH  TOA Trigonometric ratios

18. Trigonometry  The trigonometric ratios are: o Sine: opposite side over hypotenuse (SOH) o Cosine: adjacent side over hypotenuse (CAH) o Tangent: opposite side over adjacent side (TOA) Trigonometric ratios  If the lengths of a, b and c are 6, 8 and 10 what are the values of sine, cosine and tangent for angles A and B? Do you notice any patterns? What is Sin(A) and Cos(B)? What are Tan(A) and Tan(B)? o Sin(A) = a/c = 6/10 or 3/5 o Cos(A) = b/c = 8/10 or 4/5 o Tan(A) = a/b = 6/8 or 3/4 o Sin(B) = b/c = 8/10 or 4/5 o Cos(B) = a/c = 6/10 or 3/5 o Tan(B) = b/a = 8/6 or 4/3

19. Trigonometry Sine: opposite side over hypotenuse (SOH) Cosine: adjacent side over hypotenuse (CAH) Tangent: opposite side over adjacent side (TOA) Trigonometric ratios  If the lengths of a, b and c are 5, 7 and 10 what are the values of sine, cosine and tangent for A and B? Do you notice any patterns? What is Sin(A) and Cos(B)? What are Tan(A) and Tan(B)? o Sin(A) = a/c = 5/10 or 1/2 o Cos(A) = b/c = 7/10 o Tan(A) = a/b = 5/7 o Sin(B) = b/c = 7/10 o Cos(B) = a/c = 5/10 or 1/2 o Tan(B) = b/a = 7/5