Warm Up
1. What is the third angle measure in a triangle with angles measuring 65° and 43°?
Find each value. Round trigonometric ratios to the nearest hundredth and angle measures to the nearest degree.
2.
sin 73°
3.
cos 18°
4.
tan 82°
5.
sin-1 (0.34)
6.
cos-1 (0.63)
7.
tan-1 (2.75)
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Geometry/Lesson 8-5: Law of Sines and Law of Cosines
Objectives:

Use the Law of Sines and the Law of Cosines to solve triangles.
In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigonometric ratios for angle measures up to
180°. You can use a calculator to find these values.
Example 1:
Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.
1A.
tan 103
1B.
cos 165
1C.
sin 93°  1.00
cos 165°  –0.97
tan 103°  –4.33
sin 93
C.I.O.-Example 1:
1a.
tan 175
1b.
cos 92
1c.
sin 160
1
(Since “SOH CAH TOA” can only be used on right triangles, we need other tools to solve non-right triangles)
You can use the Law of Sines to solve a triangle if you are given
• two angle measures and any side length
(ASA or AAS) or
• two side lengths and a non-included angle measure (SSA).
Example 2: Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
2A.
FG
2B.
mQ
2
C.I.O.-Example 2: Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
1a.
NP
1b.
mL
2c.
mX
1d.
AC
3
The Law of Sines cannot be used to solve every triangle. If you know two side lengths and the included angle measure or if you know
all three side lengths, you cannot use the Law of Sines. Instead, you can apply the Law of Cosines.
You can use the Law of Cosines to solve a triangle if you are given
• two side lengths and the included angle measure
(SAS) or
• three side lengths (SSS).
Example 3: Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
3A.
XZ
3B.
mT
4
C.I.O.-Example 3: Find the measure. Round lengths to the nearest tenth and angle measures to the nearest degree.
3a.
DE
3b.
mK
3c.
YZ
5
Example 4:
A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC? How many
degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree.
C.I.O.-Example 4:
What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long
would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the
nearest degree.
31 m
6
Lesson Quiz: Part I
Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.
1. tan 154°
2. cos 124°
3. sin 162°
Lesson Quiz: Part II
Use ΔABC for Items 4–6. Round lengths to the nearest tenth and angle measures to the nearest degree.
4. mB = 20°, mC = 31° and b = 210. Find a.
5. a = 16, b = 10, and mC = 110°. Find c.
6. a = 20, b = 15, and c = 8.3. Find mA.
p. 573: 21-24, 28-33, 36, 37, 39, 41, 47
22) –0.56
24) –0.82
28) 4.8
30) 37
32) 12.0
36) 19.1
7