1. Use the 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. Use trigonometric ratios to solve right triangles.
Target
Use trigonometric ratios to solve right triangles.
You will…
Apply sine and cosine ratios.

3. tan(reference angle) = opposite side adjacent side
Vocabulary
tangent ratio –
tan(reference angle) =
opposite side adjacent side
sine ratio –
sin(reference angle) =
opposite side hypotenuse
cosine ratio –
cos(reference angle) =
adjacent side hypotenuse 3

4. S = sine C = cosine T = tangent
Vocabulary
S = sine C = cosine T = tangent
O = opposite H = hypotenuse A = adjacent
old indian chief Soh Cah Toa
Oscar Had A Hold On Arthur
Oh Heck Another Heap Of Apples
Some Old Horse Caught Another Horse Taking Oats Away 4

5. tan(reference angle) = opposite side adjacent side
Vocabulary
tangent ratio –
tan(reference angle) =
opposite side adjacent side
sine ratio –
sin(reference angle) =
opposite side hypotenuse
cosine ratio –
cos(reference angle) =
adjacent side hypotenuse 5

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

7. EXAMPLE 2
Find cosine ratios
Find cos U and cos W. Write each answer as a fraction and as a decimal.
SOLUTION =
adj. to U
hyp =
adj. to W
hyp
cos U
cos W = UV UW = WV UW
cos U
cos W = 18 30 = 24 30
cos U
cos W
cos U =
cos W =

8. GUIDED PRACTICE
for Example 1
Find sin X and sin Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.
ANSWER
ANSWER 8 17
sinX = 15 17
sinY = 15 25
sinX = 20 25
sinY =
0.4706
sinX
0.8824
sinY
sinX
0.6000 =
sinY
0.8000 =

9. GUIDED PRACTICE
for Example 2
Find cos X and cos Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.
ANSWER
ANSWER 15 17
cosX = 8 17
cosY = 20 25
cosX = 15 25
cosY =
0.8824
cosX
0.4706
cosY
cosX
0.8000 =
cosY
0.6000 =

10. GUIDED PRACTICE
Review
Find tan X and tan Y. Write each answer as a fraction and as a decimal. Round to four decimal places, if necessary.
ANSWER
ANSWER 8 15
tanX = 15 8
tanY = 15 20
tanX = 20 15
tanY =
0.5333
tanX
tanY
1.8750 =
tanX
0.7500 =
1.3333
tanY

11. Use a trigonometric ratio to find a hypotenuse
EXAMPLE 3
Use a trigonometric ratio to find a hypotenuse
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.
35°
x ft
11 ft
SOLUTION
First you will need to determine which trigonometric ratio is appropriate for the given information. 11 ?
tan(35)= 11 x
sin(35)= ? x
cos(35)=

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

13. Vocabulary
angle of depression – the angle formed by a horizontal line and your line of sight when looking down upon an object
angle of elevation – the angle formed by a horizontal line and your line of sight when looking up at on object 14

14. Find a hypotenuse using an angle of depression
EXAMPLE 4
Find a hypotenuse using an angle of depression
SKIING
You are skiing on a mountain with an altitude of 1200 meters. The angle of depression is 21o. About how far do you ski down the mountain?
angle of depression
SOLUTION
Determine which trigonometric ratio to use.
1200 ?
tan(21)=
1200 x
sin(21)= ? x
cos(21)=

15. Find a hypotenuse using an angle of depression
EXAMPLE 4
Find a hypotenuse using an angle of depression
SOLUTION
21°
x ft
1200 ft
opp. hyp. =
Write ratio for sine of 21o.
sin 21o
1200 x =
Substitute.
sin 21o
x sin 21o =
Multiply each side by x.
x =
1200
sin 21o
Divide each side by sin 21o x
1200
0.3584
Use a calculator to find sin 21o x
3348.2
Simplify.
ANSWER
You ski about 3348 meters down the mountain.

16. EXAMPLE 5
Find leg lengths using an angle of elevation
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.
SOLUTION
First you will need to determine which trigonometric ratio is appropriate for the given information. x y
tan(26)= x 14
sin(26)= y 14
cos(26)=

17. EXAMPLE 5
Find leg lengths using an angle of elevation
SOLUTION
Find the height.
Find the length of the base.
sin 26o =
opp. hyp.
cos 26o =
adj. hyp.
sin 26o x = 14
cos 26o y = 14
14 sin 26o
= x
14 cos 26o
= y
6.1 x
12.6 y
ANSWERS
The height is about 6.1 feet.
The length of the base is about 12.6 feet.