1. Trigonometry--The study of the properties of triangles
Trigonometry--The study of the properties of triangles. Trigonometry means angle measurement.
Trigonometric Ratio--The ratios of the measures of two sides of a right triangle.
Vocabulary

2. Concept

3. Find Sine, Cosine, and Tangent Ratios
A. Express sin L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

4. Find Sine, Cosine, and Tangent Ratios
B. Express cos L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

5. Find Sine, Cosine, and Tangent Ratios
C. Express tan L as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

6. Find Sine, Cosine, and Tangent Ratios
D. Express sin N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

7. Find Sine, Cosine, and Tangent Ratios
E. Express cos N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

8. Find Sine, Cosine, and Tangent Ratios
F. Express tan N as a fraction and as a decimal to the nearest hundredth.
Answer:
Example 1

9. A. Find sin A. A. B. C. D. A B C D
Example 1

10. B. Find cos A. A. B. C. D. A B C D
Example 1

11. C. Find tan A. A. B. C. D. A B C D
Example 1

12. D. Find sin B. A. B. C. D. A B C D
Example 1

13. E. Find cos B. A. B. C. D. A B C D
Example 1

14. F. Find tan B. A. B. C. D. A B C D
Example 1

15. The side adjacent to the 60° angle has a measure of x.
Use Special Right Triangles to Find Trigonometric Ratios
Use a special right triangle to express the cosine of 60° as a fraction and as a decimal to the nearest hundredth.
Draw and label the side lengths of a 30°-60°-90° right triangle, with x as the length of the shorter leg and 2x as the length of the hypotenuse.
The side adjacent to the 60° angle has a measure of x.
Example 2

16. Definition of cosine ratio
Use Special Right Triangles to Find Trigonometric Ratios
Definition of cosine ratio
Substitution
Simplify.
Example 2

17. Use a special right triangle to express the tangent of 60° as a fraction and as a decimal to the nearest hundredth. A. B. C. D. A B C D
Example 2

18. Estimate Measures Using Trigonometry
EXERCISING A fitness trainer sets the incline on a treadmill to 7°. 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.
Example 3

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

20. CONSTRUCTION The bottom of a handicap ramp is 15 feet from the entrance of a building. If the angle of the ramp is about 4.8°, about how high does the ramp rise off the ground to the nearest inch?
A. 1 in.
B. 11 in.
C. 16 in.
D. 15 in. A B C D
Example 3

21. Concept

22. Use a calculator to find the measure of P to the nearest tenth.
Find Angle Measures Using Inverse Trigonometric Ratios
Use a calculator to find the measure of P to the nearest tenth.
Example 4

23. Answer: So, the measure of P is approximately 46.8°.
Find Angle Measures Using Inverse Trigonometric Ratios
The measures given are those of the leg adjacent to P and the hypotenuse, so write the equation using the cosine ratio.
KEYSTROKES: [COS]
2nd ( ÷ )
ENTER
Answer: So, the measure of P is approximately 46.8°.
Example 4

24. Use a calculator to find the measure of D to the nearest tenth.
B. 48.3°
C. 55.4°
D. 57.2° A B C D
Example 4

25. Solve a Right Triangle
Solve the right triangle. Round side measures to the nearest hundredth and angle measures to the nearest degree.
Example 5

26. Step 1 Find mA by using a tangent ratio.
Solve a Right Triangle
Step 1 Find mA by using a tangent ratio.
Definition of inverse tangent
≈ mA Use a calculator.
So, the measure of A is about 30.
Example 5

27. Step 2 Find mB using complementary angles.
Solve a Right Triangle
Step 2 Find mB using complementary angles.
mA + mB = 90 Definition of complementary angles
30 + mB ≈ 90 mA ≈ 30
mB ≈ 60 Subtract 30 from each side.
So, the measure of B is about 60.
Example 5

28. Step 3 Find AB by using the Pythagorean Theorem.
Solve a Right Triangle
Step 3 Find AB by using the Pythagorean Theorem.
(AC)2 + (BC)2 = (AB)2 Pythagorean Theorem
= (AB)2 Substitution
65 = (AB)2 Simplify.
Take the positive square root of each side.
8.06 ≈ AB Use a calculator.
Example 5

29. So, the measure of AB is about 8.06.
Solve a Right Triangle
So, the measure of AB is about 8.06.
Answer: mA ≈ 30, mB ≈ 60, AB ≈ 8.06
Example 5

30. Solve the right triangle
Solve the right triangle. Round side measures to the nearest tenth and angle measures to the nearest degree.
A. mA = 36°, mB = 54°, AB = 13.6
B. mA = 54°, mB = 36°, AB = 13.6
C. mA = 36°, mB = 54°, AB = 16.3
D. mA = 54°, mB = 36°, AB = 16.3 A B C D
Example 5

31. Summary:
If you are given the angle, use sin, cos, and tan
If you want to find the angle, use sin-1, cos-1, and tan-1