This document provides examples and exercises on applying trigonometric ratios (sine, cosine, tangent) to solve problems involving right triangles. It begins by naming trigonometric ratios and defining sine, cosine and tangent. Examples are provided to calculate sine and cosine values. Follow-up exercises involve using trigonometric ratios to find missing side lengths of right triangles or calculate angle measures. Special right triangles and their related trigonometric ratio values are also discussed.

1. 7.6 Apply the Sine and Cosine Ratios
7.6
Bell Thinger
Use this diagram for Exercises 1-4.
1. Name the hypotenuse.
ANSWER XZ
2. Name the leg opposite X.
ANSWER YZ
3. Name the leg adjacent to X.
ANSWER XY
4. If XY = 17 and m X = 41 , find YZ.
ANSWER 14.78

2. 7.6

3. 7.6
Example 1
Find sin S and sin R. Write each
answer as a fraction and as a
decimal rounded to four places.
SOLUTION
sin S = opp. S
hyp
=
RT
SR
=
63
65
0.9692
sin R = opp. R
hyp
=
ST
SR
=
16
65
0.2462

4. 7.6
Guided Practice
Find sin X and sin Y. Write each answer as a fraction
and as a decimal. Round to four decimal places, if
necessary.
1.
ANSWER
8 or 0.4706, 15 or 0.8824
17
17

5. 7.6
Guided Practice
Find sin X and sin Y. Write each answer as a fraction
and as a decimal. Round to four decimal places, if
necessary.
2.
ANSWER
3 or 0.6,
5
4 or
0.8
5

6. 7.6
Example 2
Find cos U and cos W. Write each answer
as a fraction and as a decimal.
SOLUTION
cos U = adj. to U =
hyp
UV =
UW
18 = 3 = 0.6000
30
5
W
cos W = adj. to
=
hyp
WV =
UW
24 = 4 = 0.8000
30
5

7. 7.6
Example 3
DOG RUN You want to string cable to make a dog
run from two corners of a building, as shown in the
diagram. Write and solve a proportion using a
trigonometric ratio to approximate the length of cable
you will need.

8. 7.6
Example 3
SOLUTION
sin 35 =
opp.
hyp.
Write ratio for sine of 35o.
sin 35° =
11
x
Substitute.
x sin 35° = 11
x =
x
x
Multiply each side by x.
11
sin 35°
11
0.5736
Divide each side by sin 35o.
19.2
Simplify.
Use a calculator to find sin 35o.
You will need a little more than 19 feet of cable.

9. 7.6
Guided Practice
In Exercises 3 and 4, find cos R and cos S. Write each
answer as a decimal. Round to four decimal places, if
necessary.
3.
ANSWER
0.6, 0.8

10. 7.6
Guided Practice
In Exercises 3 and 4, find cos R and cos S. Write each
answer as a decimal. Round to four decimal places, if
necessary.
4.
ANSWER
0.8824, 0.4706

11. 7.6
Guided Practice
5. In Example 3, use the cosine ratio to find the
length of the other leg of the triangle formed.
ANSWER
about 15.7 ft

12. 7.6

13. 7.6
Example 4
SKIING You are skiing on a mountain with an
altitude of 1200 meters. The angle of depression is
21 . About how far do you ski down the mountain?

14. 7.6
Example 4
SOLUTION
sin 21 =
opp.
hyp.
Write ratio for sine of 21o.
sin 21° =
1200
x
Substitute.
x sin 21° = 1200
x =
1200
sin 21°
Multiply each side by x.
Divide each side by sin 21o
x
1200
0.3584
Use a calculator to find sin 21o
x
3348.2
Simplify.
You ski about 3348 meters down the mountain.

15. 7.6
Example 5
SKATEBOARD RAMP You want to build a
skateboard ramp with a length of 14 feet and an angle
of elevation of 26°. You need to find the height and
length of the base of the ramp.

16. 7.6
Example 5 SOLUTION
STEP 1 Find the height.
opp.
sin 26 =
Write ratio for sine of 26o.
hyp.
sin 26° = x
Substitute.
14
Multiply each side by 14.
14 sin 26° = x
Use a calculator to simplify.
6.1
x
The height is about 6.1 feet.
STEP 2 Find the length of the base.
adj.
cos 26° =
Write ratio for cosine of 26o.
hyp.
cos 26° = y
Substitute.
14
Multiply each side by 14.
14 cos 26° = y
Use a calculator to simplify.
12.6
y
The length of the base is about 12.6 feet.

17. 7.6
Example 6
Use a special right triangle to find the sine and cosine
of a 60o angle.
SOLUTION
Use the 30 - 60° - 90° Triangle Theorem to draw a right
triangle with side lengths of 1, 3 , and 2. Then set up
sine and cosine ratios for the 60° angle.
3
opp.
sin 60° =
=
hyp.
2
adj.
=
hyp.
1
2
cos 60° =
0.08660
= 0.5000

18. 7.6
Guided Practice
7. Use a special right triangle to find the sine and
cosine of a 30° angle.
ANSWER
1 ,
2
3
2

19. Exit Slip
7.6
Use this diagram for Exercises 1- 3.
1.
If x = 4 5 , y = 4 and z = 4 6 , find sin X, sin Y, cos X
and cos Y.
ANSWER
30
6
cos X = 6
6
sin X =
0.9129, sin Y =
0.4082, cos Y =
6
6
30
6
0.4082,
0.9129

20. Exit Slip
7.6
Use this diagram for Exercises 1- 3.
2.
If y = 10 and m
ANSWER
38.6
Y = 15 , find z to the nearest tenth.

21. Exit Slip
7.6
Use this diagram for Exercises 1- 3.
3.
If z = 12 and m
ANSWER
1.3
X = 84 , find y to the nearest tenth.

22. Exit Slip
7.6
4.
A lamppost is 11 feet tall. If the angle of elevation of
the sun is 49 , how far is the top of the lamppost
from the tip of its shadow?
ANSWER
14.6 ft

23. 7.6
Homework
Pg 495-498
#19, 21, 23, 33, 36