1. 7.5 – 7.6
Trigonometry

2. Objectives
Find trigonometric ratios (tangents, sine, and cosine) using right triangles
Solve problems using trigonometric ratios

3. Trigonometric Ratios
The word trigonometry originates from two Greek terms, trigon, which means triangle, and metron, which means measure. Thus, the study of trigonometry is the study of triangle measurements.
A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio. The three most common trigonometric ratios are sine, cosine, and tangent.

4. Trigonometric Ratios For right ∆ABC…
sin A = opposite side = a hypotenuse c
cos A = adjacent side = b                    hypotenuse c
tan A = opposite side = a                      adjacent side b A c b a C B

5. Trigonometric Ratios
To help you remember these trigonometric relationships, you can use the mnemonic device, SOH-CAH-TOA, where the first letter of each word of the trigonometric ratios is represented in the correct order. A
Sin A = Opposite side        SOH             Hypotenuse
Cos A = Adjacent side         CAH            Hypotenuse
Tan A = Opposite side    TOA                   Adjacent side c b a C B

6. Example 1:
Find sin L, cos L, tan L, sin N, cos N, and tan N. Express each ratio as a fraction and as a decimal.

7. Example 1:

8. Example 1:

9. Example 1:

10. Example 1:
Answer:

11. Your Turn:
Find sin A, cos A, tan A, sin B, cos B, and tan B. Express each ratio as a fraction and as a decimal.

12. Your Turn:
Answer:

13. Example 2a:
Use a calculator to find tan to the nearest ten thousandth.
KEYSTROKES:
TAN
ENTER
Answer:

14. Example 2b:
Use a calculator to find cos to the nearest ten thousandth.
KEYSTROKES:
COS
ENTER
Answer:

15. Your Turn:
a. Use a calculator to find sin 48° to the nearest ten thousandth.
b. Use a calculator to find cos 85° to the nearest ten thousandth.
Answer:
Answer:

16. Angles of Right Triangles
You can use a calculator or a trigonometric table to find the missing measures of a right triangle if you are given the measures of two sides of the triangle or one side and one acute angle.

17. Example 3:
EXERCISING A fitness trainer sets the incline on a treadmill to The walking surface is 5 feet long. Approximately how many inches did the trainer raise the end of the treadmill from the floor?
Let y be the height of the treadmill from the floor in inches. The length of the treadmill is 5 feet, or 60 inches.

18. Example 3: Multiply each side by 60. Use a calculator to find y.
KEYSTROKES:
SIN
ENTER
Answer: The treadmill is about 7.3 inches high.

19. Your Turn:
CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about how high does the ramp rise off the ground to the nearest inch?
Answer: about 15 in.

20. Example 4:
COORDINATE GEOMETRY Find mX in right XYZ for X(–2, 8), Y(–6, 4), and Z(–3, 1).

21. Example 4:
Explore You know the coordinates of the vertices of a right triangle and that is the right angle. You need to find the measures of one of the angles.
Plan Use the Distance Formula to find the measure of each side. Then use one of the trigonometric ratios to write an equation. Use the inverse to find
Solve or

22. Example 4: or or

23. Example 4:
Use the cosine ratio.
Simplify.
Solve for x.

24. Example 4: Use a calculator to find KEYSTROKES: 4 5
ENTER )
Examine Use the sine ratio to check the answer.
Simplify.

25. Example 4: KEYSTROKES: 3 5 Answer: The measure of is about 36.9. 2ND
ENTER )
Answer: The measure of is about 36.9.

26. Your Turn:
COORDINATE GEOMETRY
Answer: about 56.3

27. Assignment
Geometry: Workbook Pgs. 136 – 137 & 139 – #1 – 24 & #1 – 25