03/21/22 03/21/22 10.4: The Law of Sines 10.4: The Law of Sines 10.4: The Law of Sines 10.4: The Law of Sines Expectation: G1.3.2: Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle θ using the formula Area = (1/2) a b sin θ .
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04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
10.4: The Law of Sines10.4: The Law of Sines
Expectation:
G1.3.2: Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle θ using the
formula Area = (1/2) a b sin θ .
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
The Law of SinesThe Law of Sines
For any triangle ΔABC, with sides of a, b, and c opposite ∠A, ∠B and ∠C respectively:
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
3 Cases of The Law of Sines3 Cases of The Law of Sines
1. Acute Triangles
2. Right Triangles
3. Obtuse Triangles
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Proving the Law of SinesProving the Law of Sines
Given: ΔABC is acute
Prove: Sin A Sin B Sin Ca cb
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Acute Case: Law of SinesAcute Case: Law of Sines
A
B
C
a
b
c
h
x
y
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
Acute Case: Law of SinesAcute Case: Law of Sines
A
B
C
a
b
cg
t u
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
Solve the triangle below (calculate all Solve the triangle below (calculate all unknown measures).unknown measures).
74°
2015
A b
B
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Solve the triangle below.Solve the triangle below.
60° 42°5A B
C
b a
In the figure below, what is the In the figure below, what is the measure of ∠α, rounded to the measure of ∠α, rounded to the
nearest whole degree?nearest whole degree?a.a. 20°20°
b.b. 55°55°
c.c. 70°70°
d.d. 75°75°
e.e. 110°110°
04/19/2304/19/23 The Law of SinesThe Law of Sines
∠α
55°
10
11.47
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
Two cars leave the same house at the same time and each is 40 miles away after 1 hour. Car A traveled in the direction of 60° north of east. Car B traveled in the direction of 25° west of north. How far apart (from each other) are the two cars?
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Solve the triangle below. Solve the triangle below.
CA
30°
7
B
c
12
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Solve the triangle below. Solve the triangle below.
A
B
7
30°
C12
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What Happened?What Happened?
The Ambiguous Case:A triangle with an angle and 2 nonincluded
sides given (SSA) may have 2 different solutions.
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SsA ConditionSsA Condition
SSA is not a problem if the given angle is SSA is not a problem if the given angle is opposite the longer of the 2 given sides. opposite the longer of the 2 given sides. This is called SsA. A triangle to be solved is This is called SsA. A triangle to be solved is not ambiguous if it satisfies SsA.not ambiguous if it satisfies SsA.
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SsASsA5 cm
12 cm
110°This is not an ambiguous case!
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SsASsA
12 cm
5 cm
18°
This is an ambiguous case!
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What to Do? What to Do?What to Do? What to Do?
If the case is ambiguous, use sin α = sin (180 - α) to
determine both possible angle combinations.
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Solve the triangle below. Solve the triangle below.
30°
7
C
B
A
c
12
Yana is planning a triangular garden. He Yana is planning a triangular garden. He wants to put a fence around it. The wants to put a fence around it. The length of one side is 30 feet. If the length of one side is 30 feet. If the angles at each end of this side are 44° angles at each end of this side are 44° and 58°, find the length of fencing and 58°, find the length of fencing needed to enclose the garden.needed to enclose the garden.
04/19/2304/19/23 8-5: The Law of Sines8-5: The Law of Sines
04/19/2304/19/23 10.4: The Law of Sines10.4: The Law of Sines
Assignment:Assignment:
Pages 658 – 661,
# 13-21 (odds), 33-35, 51-56 (all).
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