Over Lesson 8 –5
The top of a signal tower is 250 feet above sea level. The angle of depression from the top of the tower to a passing ship is 19 ° . How far is the foot of the tower from the ship?
A.
about 81.4 ft
B.
about 236.4 ft
C.
about 726 ft
D.
about 804 ft

You used trigonometric ratios to solve right triangles.
• Use the Law of Sines to solve triangles.
• Use the Law of Cosines to solve triangles.

• Law of Sines
• Law of Cosines

Law of Sines (AAS or ASA)
Find p. Round to the nearest tenth.
We are given measures of two angles and a nonincluded side, so use the Law of Sines to write a proportion.

Law of Sines (AAS or ASA)
Law of Sines
Cross Products Property
Divide each side by sin
Use a calculator.
Answer: p ≈ 4.8

Find c to the nearest tenth.
A.
4.6
B.
29.9
C.
7.8
D.
8.5

Law of Sines (ASA)
Find x. Round to the nearest tenth.
6
57
° x

6 sin 50 =
4.8 = x x
Law of Sines (ASA) sin 73
Law of Sines m

B = 50, m

C = 73, c = 6
Cross Products Property
Divide each side by sin 73.
Use a calculator.
Answer: x ≈ 4.8

Find x. Round to the nearest degree.
A.
8
B.
10 x
C.
12
D.
14
43
°

Law of Cosines (SAS)
Find x. Round to the nearest tenth.
Use the Law of Cosines since the measures of two sides and the included angle are known.

Law of Cosines (SAS)
Law of Cosines
Simplify.
Take the square root of each side.
Use a calculator.
Answer: x ≈ 18.9

Find r if s = 15, t = 32, and m

R = 40. Round to the nearest tenth.
A.
25.1
B.
44.5
C.
22.7
D.
21.1

Law of Cosines (SSS)
Find m

L. Round to the nearest degree.
Law of Cosines
Simplify.

Law of Cosines (SSS)
Subtract 754 from each side.
Divide each side by –270.
Solve for L .
Use a calculator.
Answer: m

L ≈ 49

Find m

P. Round to the nearest degree.
A.
44 °
B.
51 °
C.
56 °
D.
69 °

Indirect Measurement
AIRCRAFT From the diagram of the plane shown, determine the approximate width of each wing. Round to the nearest tenth meter.

Indirect Measurement
Use the Law of Sines to find KJ.
Law of Sines
Cross products

Indirect Measurement
Divide each side by sin .
Simplify.
Answer: The width of each wing is about 16.9 meters.

The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Round to the nearest tenth.
A.
93.5 in.
B.
103.5 in.
C.
96.7 in.
D.
88.8 in.

Solve a Triangle
Solve triangle PQR. Round to the nearest degree.
Since the measures of three sides are given (SSS), use the Law of
Cosines to find m

P .
p 2 = r 2 + q 2 – 2 pq cos P Law of Cosines
8 2 = 9 2 + 7 2 – 2(9)(7) cos P p = 8, r = 9, and q = 7

Solve a Triangle
64 = 130 – 126 cos P
–66 = –126 cos P
Simplify.
Subtract 130 from each side.
Divide each side by –126.
Use the inverse cosine ratio.
Use a calculator.

Solve a Triangle
Use the Law of Sines to find m

Q .
Law of Sines m

P ≈ 58, p = 8, q = 7
Multiply each side by 7.
Use the inverse sine ratio.
Use a calculator.

Solve a Triangle
By the Triangle Angle Sum Theorem, m

R ≈ 180 – (58 + 48) or 74.
Answer: Therefore, m

P ≈ 58; m

Q ≈ 48 and m

R ≈ 74.

Solve ΔRST. Round to the nearest degree.
A.
m

R = 82, m

S = 58, m

T = 40
B.
m

R = 58, m

S = 82, m

T = 40
C.
m

R = 82, m

S = 40, m

T = 58
D.
m

R = 40, m

S = 58, m

T = 82