1. Sect. 9.5 Trigonometric Ratios Goal 1 Finding Trigonometric Ratios Goal 2 Using Trigonometric Ratios in Real Life

2. Finding Trigonometric Ratios Adjacent side Opposite side  hypotenuse Adjacent, Opposite Side and Hypotenuse of a Right Angle Triangle. We can also name the sides in relation to an acute angle  (Theta)

3. Finding Trigonometric Ratios Trigonometric Ratios Ratios which compare the lengths of the sides of a right triangleRatiosright triangle the common ratios are tangent, sine, and cosine. tangentsinecosine

4. Here is an easy way to remember these relationships for trig functions and the right triangle. Just write down this mnemonic: Finding Trigonometric Ratios SOH - CAH - TOA It is pronounced "so - ka - toe - ah". - The SOH stands for "Sine of an angle is Opposite over Hypotenuse." - The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse." - The TOA stands for "Tangent of an angle is Opposite over Adjacent."

5. Finding Trigonometric Ratios Example 1. The sides of a right triangle are in the ratio 3:4:5, as shown. Name and evaluate three trigonometric functions of angle  Solution:

6. Using Trigonometric Ratios in Real Life In a right triangle, sin θ =. Sketch the triangle, place the ratio numbers, and evaluate the remaining functions of θ. Example 2

7. Finding Trigonometric Ratios Use a calculator to approximate the sine, cosine and tangent of 54 . Example 3

8. Finding Trigonometric Ratios Trig functions for the 45  - 45  - 90  Triangle

9. Finding Trigonometric Ratios Trig functions for the 30  - 60  - 90  Triangle

10. Using Trigonometric Ratios in Real Life Example 4 4 7 In the figure, find sin .

11. In right  ABC, hypotenuse AB=15 and angle A=35º. Find leg BC to the nearest tenth. Using Trigonometric Ratios in Real Life Example 5:

12. Trigonometry is used typically to calculate things that we cannot measure. To measure the height h of a flagpole, we could measure a distance of, say, 100 feet from its base. From that point P, we could then measure the angle required to sight the top. If that angle turns out to be 37°, then Using Trigonometric Ratios in Real Life Example 7. Indirect measurement.

13. The angle of elevation is always measured from the ground up. Think of it like an elevator that only goes up. It is always INSIDE the triangle. In the diagram at the left, x marks the angle of elevation of the top of the tree as seen from a point on the ground. You can think of the angle of elevation in relation to the movement of your eyes. You are looking straight ahead and you must raise (elevate) your eyes to see the top of the tree. Angle of Elevation Using Trigonometric Ratios in Real Life

14. Angle of Depression In the diagram, x marks the angle of depression of a boat at sea from the top of a lighthouse. You can think of the angle of depression in relation to the movement of your eyes. You are standing at the top of the lighthouse and you are looking straight ahead. You must lower (depress) your eyes to see the boat in the water. Using Trigonometric Ratios in Real Life

15. Example 9: The angle of elevation between a point on the ground and the top of a flag pole is 30 degrees. If the point on the ground is 100 ft from the bottom of the pole, how tall is the flag pole? Using Trigonometric Ratios in Real Life

16. Find the height of the tree. Using Trigonometric Ratios in Real Life Example 10:

17. Using Trigonometric Ratios in Real Life Example 11: Find the sine, cosine, and tangent of  B. Then find the measure of  B.  B = 30.96 

18. Homework 9.5 10-38 even, 39-42 all