Q: Using the Law of Cosines, calculate the length of side $$c in a $$triangle where $$a = 5$$, $$b = 8$$, and $$C = 45^ ext{o}.$$

$$c = \sqrt{5^2$ + 8^2$ - 2 \times 5 \times 8 \times \cos 45^ ext{o}}$$

Q: A surveyor needs to find the distance between two points, $$A$$ and $$B$$, which are separated by a river. She measures $$AC = 120 m$$, $$BC = 150 m$$, and the angle $$C = 70^ ext{o}. $$What is the distance $$AB$$?

$$AB^2$$ = $$120^2$$ + $$150^2 - 2$$ \times 120 \times 150 \times \cos 70^ ext{o}$$

Q: If $$a = 10$$, $$b = 6$$, and $$C = 120^ ext{o}$$, what is the value of $$c$$ using the Law of Cosines?

$$c^2$$ = $$10^2$$ + $$6^2 - 2$$ \times 10 \times 6 \times \cos 120^ ext{o}$$

Q: A construction engineer needs to calculate the length of a support beam across a triangular roof. The sides adjacent to the beam are $$a = 14 m $$and $$b = 20 m$$, with the angle between them $$C = 75^ ext{o}. $$What is the length of the beam ($$c$$)?

$$c^2$$ = $$14^2$$ + $$20^2 - 2$$ \times 14 \times 20 \times \cos 75^ ext{o}$$

Question 1

Given a triangle with sides $$a = 7$$, $$b = 9$$, and angle $$C = 60^	ext{o}$$, which formula would you use to find the length of side $$c$$?

Accepted Answer

The Law of Cosines: $$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$ \cos C$$

Question 2

Using the Law of Cosines, calculate the length of side $$c in a $$triangle where $$a = 5$$, $$b = 8$$, and $$C = 45^	ext{o}.$$

Accepted Answer

$$c = \sqrt{5^2$ + 8^2$ - 2 	imes 5 	imes 8 	imes \cos 45^	ext{o}}$$

Question 3

A surveyor needs to find the distance between two points, $$A$$ and $$B$$, which are separated by a river. She measures $$AC = 120 m$$, $$BC = 150 m$$, and the angle $$C = 70^	ext{o}. $$What is the distance $$AB$$?

Accepted Answer

$$AB^2$$ = $$120^2$$ + $$150^2 - 2$$ 	imes 120 	imes 150 	imes \cos 70^	ext{o}$$

Question 4

Which of the following is a correct statement of the Law of Cosines for any triangle $$ABC$$?

Accepted Answer

$$a^2$$ = $$b^2$$ + $$c^2 - 2bc$$ \cos A$$

Question 5

If $$a = 10$$, $$b = 6$$, and $$C = 120^	ext{o}$$, what is the value of $$c$$ using the Law of Cosines?

Accepted Answer

$$c^2$$ = $$10^2$$ + $$6^2 - 2$$ 	imes 10 	imes 6 	imes \cos 120^	ext{o}$$

Question 6

A triangle has sides $$a = 8$$, $$b = 15$$, and $$c = 17. $$Which formula would you use to find angle $$C$$?

Accepted Answer

$$\cos C = \frac{a^2$ + b^2$ - c^2$}{2ab}$$

Question 7

In a triangle, if $$a = 12$$, $$b = 9$$, and $$c = 7$$, what is the value of angle $$A$$ using the Law of Cosines?

Accepted Answer

$$\cos A = \frac{b^2$ + c^2$ - a^2$}{2bc}$$

Question 8

A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 10. $$Which of the following is the correct way to find angle $$B$$?

Accepted Answer

$$\cos B = \frac{a^2$ + c^2$ - b^2$}{2ac}$$

Question 9

A construction engineer needs to calculate the length of a support beam across a triangular roof. The sides adjacent to the beam are $$a = 14 m $$and $$b = 20 m$$, with the angle between them $$C = 75^	ext{o}. $$What is the length of the beam ($$c$$)?

Accepted Answer

$$c^2$$ = $$14^2$$ + $$20^2 - 2$$ 	imes 14 	imes 20 	imes \cos 75^	ext{o}$$

Question 10

Which scenario best illustrates the use of the Law of Cosines in real life?

Accepted Answer

Calculating the distance between two points on a map when the angle between them and their distances from a third point are known.

Question 11

If $$a = 9$$, $$b = 12$$, and $$C = 90^	ext{o}$$, what is the value of $$c$$?

Accepted Answer

$$c^2$$ = $$9^2$$ + $$12^2 - 2$$ 	imes 9 	imes 12 	imes \cos 90^	ext{o}$$

Question 12

A triangle has sides $$a = 6$$, $$b = 8$$, and $$c = 10. $$What is the measure of angle $$C$$?

Accepted Answer

$$\cos C = \frac{a^2$ + b^2$ - c^2$}{2ab}$$

The Law of Cosines in Trigonometry

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