Download 1. A right triangle has legs of 8 centimeters and... centimeters. Solve the triangle completely.

9.7 Warmup
1. A right triangle has legs of 8 centimeters and 13
centimeters. Solve the triangle completely.
2. A right triangle has a leg length of 7 in. and a
hypotenuse length of 14 in. Solve the triangle completely.
3. A right triangle has legs that both measure to be 10
meters. Solve the triangle completely.
April 1, 2016
9.7 Law of Sines and Cosines
1
Geometry
9.7 Law of Sines and Cosines
9.7 Essential Question
When do you use the Law of Sines and the Law of
Cosines?
April 1, 2016
9.7 Law of Sines and Cosines
3
What if you have a triangle that is not
a right triangle?
So far, you have used trigonometric ratios to solve right
triangles. In this lesson, you will learn how to solve obtuse
and acute triangles too.
When the triangle is obtuse, you may need to find a
trigonometric ratio for an obtuse angle.
April 1, 2016
9.7 Law of Sines and Cosines
4
Example 1
Use a calculator to find each trigonometric ratio.
Round your answer to four decimal places.
a. tan 150°
April 1, 2016
b. sin 120°
9.7 Law of Sines and Cosines
c. cos 95°
5
Area of a Triangle
April 1, 2016
9.7 Law of Sines and Cosines
6
Example 2
Find the area of the triangle. Round your answer to the
nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
7
Your Turn
Find the area of Δ ABC with the given side lengths and
included angle. Round your answer to the nearest tenth.
C = 29°, a = 38, b = 31
April 1, 2016
9.7 Law of Sines and Cosines
8
Law of Sines
April 1, 2016
9.7 Law of Sines and Cosines
9
When do you use the Law of Sines?
There are two cases for using the Law of Sines:
Given 2 angles and
any side… AAS or ASA
Find the side opposite
an angle.
April 1, 2016
OR
Given 2 sides and
1 opposite angle…
Find the angle
opposite a side.
9.7 Law of Sines and Cosines
SSA
10
Example 3
Solve the triangle. Round decimal answers to the nearest tenth.
Use the Law of Sines to find 𝑚∠B.
sin 𝐵 sin 𝐴
=
𝑏
𝑎
sin 𝐵 sin 115°
=
11
20
11 𝑠𝑖𝑛115°
𝑠𝑖𝑛𝐵 =
20
𝑚∠𝐵 ≈ 29.9°
April 1, 2016
By the Triangle Sum Theorem,
𝑚∠𝐶 ≈ 180° − 115° − 29.9°
𝑚∠𝐶 ≈ 35.1°
9.7 Law of Sines and Cosines
11
Example 3 (continued)
Solve the triangle. Round decimal answers to the nearest tenth.
Use the Law of Sines again to find the remaining side length, c, of the triangle.
𝑐
𝑎
=
sin 𝐶 sin 𝐴
𝑐
20
=
sin 35.1° sin 115°
20 sin 35.1°
𝑐=
𝑠𝑖𝑛115°
In ΔABC, 𝑚∠𝐵 ≈ 29.9°,
𝑚∠𝐶 ≈ 35.1°, and 𝑐 ≈ 12.7.
𝑐 ≈ 12.7
April 1, 2016
9.7 Law of Sines and Cosines
12
Your Turn
Solve the triangle. Round decimal answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
13
Example 4
Solve the triangle. Round decimal answers to the nearest tenth.
By the Triangle Sum Theorem, 𝑚∠𝐴 = 180° − 107° − 25° = 48°
By the Law of Sines, you can write
𝑎
15
=
sin 48° sin 25°
𝑎=
15 sin 48°
sin 25°
𝑎 ≈ 26.4
April 1, 2016
Write two equations,
each with one variable.
𝑎
sin 48°
=
15
sin 25°
=
𝑐
15
=
sin 107° sin 25°
𝑐=
𝑐
sin 107°
In ΔABC,
𝑚∠𝐴 = 48°
𝑎 ≈ 26.4
𝑐 ≈ 33.9
15 sin 107°
sin 25°
𝑐 ≈ 33.9
9.7 Law of Sines and Cosines
14
Your Turn
Solve the triangle. Round decimal answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
15
Example 5
A surveyor makes the measurements shown to determine the
length of a bridge to be built across a small lake from the North
Picnic Area to the South Picnic Area. Find the length of the
bridge.
In the diagram, the bridge will be the length of c.
𝑐
𝑏
By the Law of Sines,
=
sin 𝐶
sin 𝐵
By the Triangle Sum Theorem, 𝑚∠𝐵 = 180° − 71° − 60° = 49°
𝑐
sin 60°
𝑐=
=
150
sin 49°
150 sin 60°
sin 49°
April 1, 2016
The length of the bridge
will be about 172.1 meters.
𝑐 ≈ 172.1
9.7 Law of Sines and Cosines
16
Law of Cosines
April 1, 2016
9.7 Law of Sines and Cosines
17
When do you use the Law of Cosines?
There are two cases for using the Law of Cosines:
Given 2 sides and
1 included angle…
Find the third side
April 1, 2016
or
9.7 Law of Sines and Cosines
Given 3 sides…
Find any angle
18
Example 6
Solve the triangle. Round decimal
answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
19
Example 6 (continued)
Solve the triangle. Round decimal
answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
20
Your Turn
Solve the triangle. Round decimal
answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
21
Example 7
Solve the triangle. Round decimal answers to
the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
22
Example 7 (continued)
Solve the triangle. Round decimal
answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
23
Your Turn
Solve the triangle. Round decimal
answers to the nearest tenth.
April 1, 2016
9.7 Law of Sines and Cosines
24
Homework
April 1, 2016
9.7 Law of Sines and Cosines
25