unit 5.6
TRANSCRIPT
Copyright © 2011 Pearson, Inc.
5.6Law of Cosines
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What you’ll learn about
Deriving the Law of Cosines Solving Triangles (SAS, SSS) Triangle Area and Heron’s Formula Applications
… and whyThe Law of Cosines is an important extension of the Pythagorean theorem, with many applications.
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Law of Cosines
Let VABC be any triangle with sides and angles
labeled in the usual way. Then
a2 b2 c2 2bccos A
b2 a2 c2 2accos B
c2 a2 b2 2abcosC
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Example Solving a Triangle (SAS)
Solve VABC given that a 10, b 4 and C 25o.
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Example Solving a Triangle (SAS)
Use the Law of Cosines to find side c:
c2 a2 b2 2abcosC
c2 16 100 2(4)(10)cos25o
c 6.6
Solve VABC given that a 10, b 4 and C 25o.
Use the Law of Cosines again:
102 16 43.56 2(4)(6.6)cos A
cos A 0.7659
A 140o
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Example Solving a Triangle (SAS)
Solve VABC given that a 10, b 4 and C 25o.
Now find (sum of the angles in a triangle = 180º):
B 180o 140o 25o 15o
The six parts of the triangle are:
A 140o a 10,
B 15o b 4,
C 25o c 6.6
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Area of a Triangle
1 1 1
Area sin sin sin2 2 2
bc A ac B ab C
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Heron’s Formula
Let a, b, and c be the sides of VABC, and let s denote
the semiperimeter (a b c) / 2. Then the area of
VABC is given by
Area s s a s b s c .
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Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
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Example Using Heron’s Formula
Find the area of a triangle with sides 10, 12, 14.
Compute s: s (10 12 14) / 2 18.
Use Heron's Formula:
A 18 18 10 18 12 18 14 = 3456
=24 6 58.8
The area is approximately 58.8 square units.
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Quick Review
Find an angle between 0o and 180o that is a solution
to the equation.
1. cos A 4 / 5
2. cos A -0.25
Solve the equation (in terms of x and y) for
(a) cos A and (b) A, 0 A 180o.
3. 72 x2 y2 2xycos A
4. y2 x2 4 4xcos A
5. Find a quadratic polynomial with real coefficients
that has no real zeros.
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Quick Review
Find an angle between 0o and 180o that is a solution
to the equation.
1. cos A 4 / 5 36.87o
2. cos A 0.25 104.48o
Solve the equation (in terms of x and y) for
(a) cos A and (b) A, 0 A 180o.
3. 72 x2 y2 2xycos A
(a) 49 x2 y2
2xy (b) cos-1 49 x2 y2
2xy
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Quick Review
Solve the equation (in terms of x and y) for
(a) cos A and (b) A, 0 A 180o.
4. y2 x2 4 4xcos A
(a) y2 x2 4
4x (b) cos-1 y2 x2 4
4x
5. Find a quadratic polynomial with real coefficients
that has no real zeros.
One answer: x2 2
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Chapter Test
1. Prove the identity cos3x 4cos3 x 3cos x.
2. Write the expression in terms of sin x and cos x.
cos2 2x sin2x
3. Find the general solution without using a calculator.
2cos2x 1
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Chapter Test
4. Solve the equation graphically. Find all solutions
in the interval [0,2 ). sin4 x x2 2
5. Find all solutions in the interval [0,2 ) without
using a calculator. sin2 x 2sin x 3 0
6. Solve the inequality. Use any method, but give
exact answers. 2cos x 1 for 0 x 27. Solve VABC, given A 79o, B 33o, and a 7.
8. Find the area of VABC, given a 3, b 5, and c 6.
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Chapter Test
9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?
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Chapter Test
10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.
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Chapter Test Solutions
1. Prove the identity cos3x 4cos3 x 3cos x.
cos3x cos(2x x) cos2xcos x sin2xsin x
cos2 x sin2 x cos x 2sin xcos x sin x
cos3 x 3cos xsin2 x
cos3 x 3cos x 1 cos2 x 4cos3 x 3cos x.
2. Write the expression in terms of sin x and cos x.
cos2 2x sin2x 1 4sin2 xcos2 x 2cos xsin x
3. Find the general solution without using a calculator.
2cos2x 1 6
2n , 56
2n
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Chapter Test Solutions
4. Solve the equation graphically. Find all solutions
in the interval [0,2 ). sin4 x x2 2 x 1.15
5. Find all solutions in the interval [0,2 ) without
using a calculator. sin2 x 2sin x 3 0 32
6. Solve the inequality. Use any method, but give
exact answers. 2cos x 1 for 0 x 2 3
,53
7. Solve VABC, given A 79o, B 33o, and a 7.
C 68o, b 3.88, c 6.61
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Chapter Test Solutions
9. A hot-air balloon is seen over Tucson, Arizona, simultaneously by two observers at points A and B that are 1.75 mi apart on level ground and in line with the balloon. The angles of elevation are as shown here. How high above ground is the balloon?
≈ 0.6 mi
8. Find the area of VABC, given a 3, b 5, and c 6.
7.5
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Chapter Test Solutions
10. A wheel of cheese in the shape of a right circular cylinder is 18 cm in diameter and 5 cm thick. If a wedge of cheese with a central angle of 15º is cut from the wheel, find the volume of the cheese wedge.
405π/24 ≈ 53.01