Transcript Apply Sine and Cosine Ratios

Apply Sine and Cosine
Ratios
5.3 (M2)
Vocabulary
Sine and Cosine ratios: trig. Ratios for acute
angles with legs and hypotenuse
legoppositeA BC
sin A 

hypotenuse
AB
legadjacent A AC
cos A 

hypotenuse
AB
B
C
A
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
sin S = opp. S
hyp
=
RT
SR
=
63
65
0.9692
sin R = opp. R
hyp
=
ST
SR
=
16
65
0.2462
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
8 or 0.4706,
17
ANSWER
3 or 0.6,
5
15 or 0.8824
17
4 or 0.8
5
EXAMPLE 2
Find cosine ratios
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
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.
EXAMPLE 3
Use a trigonometric ratio to find a hypotenuse
SOLUTION
sin 35o =
opp.
hyp.
sin 35o =
11
x
Substitute.
x sin 35o =
11
Multiply each side by x.
x =
x
x
11
sin 35o
11
0.5736
19.2
Write ratio for sine of 35o.
Divide each side by sin 35o.
Use a calculator to find sin 35o.
Simplify.
ANSWER You will need a little more than 19 feet of cable.
GUIDED PRACTICE
for Examples 2 and 3
In Exercises 3 and 4, find cos R and cos S. Write each
answer as a decimal. Round to four decimal places, if
necessary.
ANSWER
0.6, 0.8
ANSWER
0.8824, 0.4706
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
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?
EXAMPLE 4
Find a hypotenuse using an angle of depression
SOLUTION
sin 21 =
opp.
hyp.
Write ratio for sine of 21o.
sin 21o =
1200
x
Substitute.
o
x sin 21o = 1200
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.
GUIDED PRACTICE
6.
for Example 4
WHAT IF?
Suppose the angle of depression in Example 4 is 28°.
About how far would you ski?
ANSWER
about 2556 m
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.
EXAMPLE 5
Find leg lengths using an angle of elevation
SOLUTION
STEP 1
Find the height.
opp.
sin 26o =
hyp.
x
14
sin 26o =
14 sin 26o = x
6.1
ANSWER
x
Write ratio for sine of 26o.
Substitute.
Multiply each side by 14.
Use a calculator to simplify.
The height is about 6.1 feet.
EXAMPLE 5
STEP 2
Find leg lengths using an angle of elevation
Find the length of the base.
adj.
cos 26o =
hyp.
y
14
cos 26o =
14 cos 26o = y
12.6
ANSWER
y
Write ratio for cosine of 26o.
Substitute.
Multiply each side by 14.
Use a calculator to simplify.
The length of the base is about 12.6 feet.
EXAMPLE 6 Use a special right triangle to find a sine and cosine
Use a special right triangle to find the sine and cosine
of a 60o angle.
SOLUTION
Use the 30o - 60o - 90o Triangle Theorem to draw a right
triangle with side lengths of 1, 3 , and 2. Then set up
sine and cosine ratios for the 60o angle.
o
sin 60
opp.
=
=
hyp.
3
2
adj.
=
hyp.
1
2
cos 60o =
0.08660
= 0.5000