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
Law of Sines
57
°
x
mB = 50, mC = 73, c = 6
6 sin 50 = x sin 73
Cross Products Property
Divide each side by sin 73.
4.8 = x
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
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.
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.
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.
p2 = r2 + q2 – 2pq cos P
Law of Cosines
82 = 92 + 72 – 2(9)(7) cos P
p = 8, r = 9, and q = 7
64 = 130 – 126 cos P
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.
Use the inverse
sine ratio.
Use a calculator.
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
SAS
AAS
ASA
p. 592
8-6 Assignment day 2
p. 592, 31-42