1. Objectives Find the sine, cosine, and tangent of an acute angle.
Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems.

2. Vocabulary
trigonometric ratio
sine
cosine
tangent

3. _________________– a ratio of two sides of a right triangle.

5. In trigonometry, the letter of the vertex of the angle is often used to represent the measure of that angle. For example, the sine of A is written as sin A.
Writing Math

6. Example 1A: Finding Trigonometric Ratios
Write the trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.
sin J
cos J
tan K

7. Example 1b
Write the trigonometric ratio as a fraction and as a decimal rounded to
the nearest hundredth.
cos A
tan B
sin B

8. Example 2a
Use a special right triangle to write cos 30° as a fraction.
Draw and label a 30º-60º-90º ∆.

9. Example 2b
Use a special right triangle to write tan 45° as a fraction.
Draw and label a 45º-45º-90º ∆.

10. Example 3A: Calculating Trigonometric Ratios
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
sin 52°
Be sure your calculator is in degree mode, not radian mode.
Caution!

11. Example 3 Continued
Use your calculator to find the trigonometric ratio. Round to the nearest hundredth.
3b. cos 19°
3c. tan 65°
3d. sin 62°

12. Example 4A: Using Trigonometric Ratios to Find Lengths
Find the length. Round to the nearest hundredth. BC

13. Caution!
Do not round until the final step of your answer. Use the values of the trigonometric ratios provided by your calculator.

14. Example 4B: Using Trigonometric Ratios to Find Lengths
Find the length. Round to the nearest hundredth. QR

15. Example 4C: Using Trigonometric Ratios to Find Lengths
Find the length. Round to the nearest hundredth. FD

16. Example 4d
Find the length. Round to the nearest hundredth. JL

17. Example 5a: Problem-Solving Application
The Pilatusbahn in Switzerland is the world’s steepest cog railway. Its steepest section makes an angle of about 25.6º with the horizontal and rises about 0.9 km. To the nearest hundredth of a kilometer, how long is this section of the railway track?

18. Understand the Problem
Example 5a Continued 1
Understand the Problem
Make a sketch.
0.9 km

19. Example 5a Continued 2
Make a Plan
Solve 3

20. Example 5b
Find AC, the length of the ramp, to the nearest hundredth of a foot.

21. Understand the Problem
Example 5b Continued 1
Understand the Problem 2
Make a Plan

22. Example 5b Continued
Solve 3