Transcript 8-6 The Law of Sines and Law of Cosines
8-6 The Law of Sines and Law of Cosines
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
• The Law of Sines can be used to find side lengths and angle measures for any triangle (nonright triangles).
• You can use the Law of Sines to solve a triangle if you know the measures of two angles and any side (AAS or ASA) p. 588
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 (AAS).
Law of Sines Cross Products Property Divide each side by sin
Answer:
Use a calculator.
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 sin 50 =
x
sin 73 4.8 =
x
6 Law of Sines
m
B
57 = 50,
m
C
° = 73,
c x
= 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 C.
12 D.
14
° 43
x
Law of Cosines
You can use the Law of Cosines to solve a triangle if you know the measures of two sides and the included angle (SAS) p. 589
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 Simplify.
Answer:
Take the square root of each side.
Use a calculator.
x ≈ 18.9
A.
Find r if s = 15, t = 32, and m
R = 40. Round to the nearest tenth.
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.
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°
8-6 Assignment, day 1 p. 592, 12-17, 22-27
8-6 The Law of Sines and Law of Cosines, day 2
You used trigonometric ratios to solve right triangles. • Use the Law of Sines to solve triangles.
• Use the Law of Cosines to solve triangles.
Solve a Triangle
When solving
right
triangles, you can use sine, cosine, or tangent.
When solving
other
triangles, you can use the Law of Sines or the Law of Cosines, depending on what information is given AAS, ASA for sines SAS, SSS for cosines)
AIRCRAFT From the diagram of the plane shown, determine the approximate width of each wing. Round to the nearest tenth meter.
Use the Law of Sines to find KJ. Law of Sines Cross products
Cross products 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 – 2pq cos P Law of Cosines 8 2 = 9 2 + 7 2 – 2(9)(7) cos P 64 = 130 – 126 cos P p = 8, r = 9, and q = 7 Simplify.
–66 = –126 cos P Subtract 130 from each side.
Divide each side by –126.
Use the inverse cosine ratio.
Use a calculator.
Use the Law of Sines to find m Q.
Law of Sines
m
P ≈ 58, p = 8, q = 7 Multiply each side by 7.
By the Triangle Angle Sum Theorem,
m
R ≈ 180 – (58 + 48) or 74.
Use the inverse sine ratio.
Use a calculator.
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
SAS ASA AAS
p. 592
8-6 Assignment day 2
p. 592, 31-42