Q: If $$a = 8$$, $$b = 5$$, and $$C = 60^ ext{o}$$, what is the value of $$c$$ using the Law of Cosines?

$$c = \sqrt{8^2$ + 5^2$ - 2 \times 8 \times 5 \times \cos 60^ ext{o}}$$

Q: A triangle has sides $$a = 10$$, $$b = 6$$, and angle $$C = 90^ ext{o}. $$What is the length of side $$c$$?

$$c = \sqrt{10^2$ + 6^2$ - 2 \times 10 \times 6 \times \cos 90^ ext{o}}$$

Q: If $$a = 12$$, $$b = 9$$, and $$C = 45^ ext{o}$$, calculate $$c$$ using the Law of Cosines.

$$c = \sqrt{12^2$ + 9^2$ - 2 \times 12 \times 9 \times \cos 45^ ext{o}}$$

Q: A construction engineer needs to determine the length of a support beam across a triangular roof. The two sides of the triangle are $$15 ft $$and $$20 ft$$, and the angle between them is $$75^ ext{o}. $$What is the length of the beam?

$$c = \sqrt{15^2$ + 20^2$ - 2 \times 15 \times 20 \times \cos 75^ ext{o}}$$

Question 1

Given triangle ABC with sides $$a = 7$$, $$b = 9$$, and angle $$C = 120^	ext{o}$$, which law should you use to find side $$c$$ and why?

Accepted Answer

Use the Law of Cosines because you have two sides and the included angle.

Question 2

What is the Law of Cosines formula for finding side $$c in a $$triangle with sides $$a$$, $$b$$, and angle $$C$$?

Accepted Answer

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

Question 3

A surveyor needs to find the distance across a river. She measures two points on one bank, $$A$$ and $$B$$, which are $$100$$ meters apart. She then measures the angle from $$A to a $$point $$C on $$the opposite bank as $$45^	ext{o}$$ and from $$B to$$ $$C as$$ $$60^	ext{o}. $$Which law should she use to find the distance $$AC$$?

Accepted Answer

Law of Sines, because two angles and one side are known.

Question 4

If $$a = 8$$, $$b = 5$$, and $$C = 60^	ext{o}$$, what is the value of $$c$$ using the Law of Cosines?

Accepted Answer

$$c = \sqrt{8^2$ + 5^2$ - 2 	imes 8 	imes 5 	imes \cos 60^	ext{o}}$$

Question 5

Why is the Law of Cosines preferred over the Law of Sines when solving for a side given two sides and the included angle?

Accepted Answer

Because the Law of Sines requires knowing an angle opposite a known side.

Question 6

A triangle has sides $$a = 10$$, $$b = 6$$, and angle $$C = 90^	ext{o}. $$What is the length of side $$c$$?

Accepted Answer

$$c = \sqrt{10^2$ + 6^2$ - 2 	imes 10 	imes 6 	imes \cos 90^	ext{o}}$$

Question 7

Which of the following is a scenario where the Law of Sines is NOT applicable?

Accepted Answer

When you know two sides and the included angle.

Question 8

A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 9. $$Which law should be used to find the largest angle?

Accepted Answer

Law of Cosines, because all three sides are known.

Question 9

If $$a = 12$$, $$b = 9$$, and $$C = 45^	ext{o}$$, calculate $$c$$ using the Law of Cosines.

Accepted Answer

$$c = \sqrt{12^2$ + 9^2$ - 2 	imes 12 	imes 9 	imes \cos 45^	ext{o}}$$

Question 10

A construction engineer needs to determine the length of a support beam across a triangular roof. The two sides of the triangle are $$15 ft $$and $$20 ft$$, and the angle between them is $$75^	ext{o}. $$What is the length of the beam?

Accepted Answer

$$c = \sqrt{15^2$ + 20^2$ - 2 	imes 15 	imes 20 	imes \cos 75^	ext{o}}$$

Question 11

Which of the following is the correct rearrangement of the Law of Cosines to solve for angle $$C$$?

Accepted Answer

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

Question 12

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

Accepted Answer

$$\cos C = \frac{6^2$ + 8^2$ - 10^2$}{2 	imes 6 	imes 8}$$

Question 13

A pilot is flying from point $$A to $$point $$B$$ and then to point $$C. $$The distance from $$A to$$ $$B is$$ $$120 km$$, from $$B to$$ $$C is$$ $$150 km$$, and the angle at $$B is$$ $$110^	ext{o}. $$What is the direct distance from $$A to$$ $$C$$?

Accepted Answer

$$AC = \sqrt{120^2$ + 150^2$ - 2 	imes 120 	imes 150 	imes \cos 110^	ext{o}}$$

Question 14

Which of the following is NOT a valid use case for the Law of Cosines?

Accepted Answer

Finding a side when two angles and one side are known.

Solving Triangles Using the Law of Cosines

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