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

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.

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

α = 120° , b = 6 , c = 7

11.3

β = 58.7° , a = 10.6 , c = 15.7

γ = 115° , a = 18 , b = 23

34.7

α = 119° , a = 26 , b = 14

γ = 113° , b = 10 , c = 32

26.7

β = 67° , a = 49 , b = 38

α = 43.1° , a = 184.2 , b = 242.8

257.4

α = 36.6° , a = 186.2 , b = 242.2

β = 50° , a = 105 , b = 45

not possible

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 .

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

95.5°

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

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

26.9°

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

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

B = 88° , a = 4.4 , c = 5.2

C = 121° , a = 21 , b = 37

A 20.6° , B 38.4° , c 51.1

a = 13 , b = 11 , c = 15

a = 3.1 , b = 3.5 , c = 5

A 37.8° , B 43.8 , C 98.4°

a = 51 , b = 25 , c = 29

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

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

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

0.04 m 2

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

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

0.91 yd 2

Graphical

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

A triangle. One angle is 72 degrees, with opposite side = x. The other two sides are 5 and 6.5.
A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.

3.0

A triangle. One angle is 40 degrees with opposite side = 15. The other two sides are 12 and x.
A triangle. One angle is 65 degrees with opposite side = x. The other two sides are 30 and 23.

29.1

A triangle. One angle is 50 degrees with opposite side = x. The other two sides are 225 and 305.
A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

0.5

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

A triangle. Angle A is opposite a side of length 2.3. The other two sides are 1.5 and 2.5.
A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.

70.7°

A triangle. Angle A is opposite a side of length 6.8. The other two sides are 4.3 and 8.2.
A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.

77.4°

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.

For the following exercises, solve for the unknown side. Round to the nearest tenth.

A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.

25.0

A triangle. One angle is 30 degrees with opposite side unknown. The other two sides are 16 and 10.
A triangle. One angle is 22 degrees with opposite side unknown. The other two sides are 20 and 13.

9.3

A triangle. One angle is 88 degrees with opposite side = 9. Another side is 5.

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

A triangle with sides 8, 12, and 17. Angles unknown.

43.52

A triangle with sides 50, 22, and 36. Angles unknown.
A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.

1.41

A triangle with sides 8.9, 12.5, and 16.2. Angles unknown.
A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

0.14

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.

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

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.

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

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.

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.

Draw the triangle.

A triangle. One angle is 52 degrees with opposite side = x. The other two sides are 5 and 6.

Find the length of the third side.

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

A triangle. One angle is 22 degrees with opposite side = 3.4. Another side is 5.3.

7.62

Practice Key Terms 2

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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