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Explain the relationship between the Pythagorean Theorem and the Law of Cosines.

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When must you use the Law of Cosines instead of the Pythagorean Theorem?

The Law of Cosines must be used for any oblique (non-right) triangle.

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Algebraic

For the following exercises, assume α is opposite side a , β is opposite side b , and γ is opposite side c . If possible, solve each triangle for the unknown side. Round to the nearest tenth.

γ = 41.2° , a = 2.49 , b = 3.13

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α = 120° , b = 6 , c = 7

11.3

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β = 58.7° , a = 10.6 , c = 15.7

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γ = 115° , a = 18 , b = 23

34.7

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α = 119° , a = 26 , b = 14

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γ = 113° , b = 10 , c = 32

26.7

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β = 67° , a = 49 , b = 38

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α = 43.1° , a = 184.2 , b = 242.8

257.4

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α = 36.6° , a = 186.2 , b = 242.2

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β = 50° , a = 105 , b = 45

not possible

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For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

a = 42 , b = 19 , c = 30 ; find angle A .

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a = 14 ,   b = 13 ,   c = 20 ; find angle C .

95.5°

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a = 16 , b = 31 , c = 20 ; find angle B .

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a = 13 , b = 22 , c = 28 ; find angle A .

26.9°

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a = 108 , b = 132 , c = 160 ; find angle C .

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For the following exercises, solve the triangle. Round to the nearest tenth.

A = 35° , b = 8 , c = 11

B 45.9° , C 99.1° , a 6.4

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B = 88° , a = 4.4 , c = 5.2

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C = 121° , a = 21 , b = 37

A 20.6° , B 38.4° , c 51.1

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a = 13 , b = 11 , c = 15

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a = 3.1 , b = 3.5 , c = 5

A 37.8° , B 43.8 , C 98.4°

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a = 51 , b = 25 , c = 29

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For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

177.56 in 2

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Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

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a = 1 2 m , b = 1 3 m , c = 1 4 m

0.04 m 2

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a = 12.4  ft ,   b = 13.7  ft ,   c = 20.2  ft

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a = 1.6  yd ,   b = 2.6  yd ,   c = 4.1  yd

0.91 yd 2

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Graphical

For the following exercises, find the length of side x . Round to the nearest tenth.

For the following exercises, find the measurement of angle A .

Find the measure of each angle in the triangle shown in [link] . Round to the nearest tenth.

A triangle A B C. Angle A is opposite a side of length 10, angle B is opposite a side of length 12, and angle C is opposite a side of length 7.
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For the following exercises, solve for the unknown side. Round to the nearest tenth.

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

Extensions

A parallelogram has sides of length 16 units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

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The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

18.3

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The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure of the larger angle is 100°. Find the length of the shorter diagonal.

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A regular octagon is inscribed in a circle with a radius of 8 inches. (See [link] .) Find the perimeter of the octagon.

An octagon inscribed in a circle.

48.98

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A regular pentagon is inscribed in a circle of radius 12 cm. (See [link] .) Find the perimeter of the pentagon. Round to the nearest tenth of a centimeter.

A pentagon inscribed in a circle.
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For the following exercises, suppose that x 2 = 25 + 36 60 cos ( 52 ) represents the relationship of three sides of a triangle and the cosine of an angle.

Find the length of the third side.

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For the following exercises, find the area of the triangle.

Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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