Q: A surveyor measures two sides of a triangular plot of land as $$a = 120 m $$and $$b = 150 m$$, with the included angle $$C = 60^ ext{o}. $$What is the length of the third side $$c$$? Use the Law of Cosines.

$$c = \sqrt{120^2$ + 150^2$ - 2 \times 120 \times 150 \times \cos 60^ ext{o}}$$

Q: If $$a = 8$$, $$b = 15$$, and $$c = 17$$, what is the measure of angle $$C$$ opposite side $$c$$? Use the Law of Cosines.

$$C = \cos$$^{-1}$$ \left( \frac{8^2$ + 15^2$ - 17^2$}{2 \times 8 \times 15} \right)$$

Q: A ship travels 5 km east, then 7 km northeast. What is the straight-line distance from its starting point to its final position? (Assume northeast is $$45^ ext{o}$$ from east.)

$$\sqrt{5^2$ + 7^2$ - 2 \times 5 \times 7 \times \cos 135^ ext{o}}$$

Q: Given triangle $$ABC$$ with $$a = 10$$, $$b = 7$$, and $$C = 120^ ext{o}$$, what is the length of side $$c$$?

$$c = \sqrt{10^2$ + 7^2$ - 2 \times 10 \times 7 \times \cos 120^ ext{o}}$$

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

$$C = \cos$$^{-1}$$ \left( \frac{6^2$ + 8^2$ - 10^2$}{2 \times 6 \times 8} \right)$$

Q: A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 9. $$Which of the following is the correct expression for finding angle $$C$$?

$$C = \cos$$^{-1}$$ \left( \frac{5^2$ + 7^2$ - 9^2$}{2 \times 5 \times 7} \right)$$

Q: A triangle has sides $$a = 13$$, $$b = 14$$, and $$c = 15. $$What is the value of $$\cos C$$?

$$\cos C = \frac{13^2$ + 14^2$ - 15^2$}{2 \times 13 \times 14}$$

Question 1

Which formula represents the Law of Cosines for a triangle with sides $$a$$, $$b$$, $$c$$ and angle $$C$$ opposite side $$c$$?

Accepted Answer

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

Question 2

A surveyor measures two sides of a triangular plot of land as $$a = 120 m $$and $$b = 150 m$$, with the included angle $$C = 60^	ext{o}. $$What is the length of the third side $$c$$? Use the Law of Cosines.

Accepted Answer

$$c = \sqrt{120^2$ + 150^2$ - 2 	imes 120 	imes 150 	imes \cos 60^	ext{o}}$$

Question 3

If $$a = 8$$, $$b = 15$$, and $$c = 17$$, what is the measure of angle $$C$$ opposite side $$c$$? Use the Law of Cosines.

Accepted Answer

$$C = \cos$$^{-1}$$ \left( \frac{8^2$ + 15^2$ - 17^2$}{2 	imes 8 	imes 15} ight)$$

Question 4

A ship travels 5 km east, then 7 km northeast. What is the straight-line distance from its starting point to its final position? (Assume northeast is $$45^	ext{o}$$ from east.)

Accepted Answer

$$\sqrt{5^2$ + 7^2$ - 2 	imes 5 	imes 7 	imes \cos 135^	ext{o}}$$

Question 5

Which scenario is most appropriate for using the Law of Cosines instead of the Law of Sines?

Accepted Answer

When you know two sides and the included angle of a triangle.

Question 6

Given triangle $$ABC$$ with $$a = 10$$, $$b = 7$$, and $$C = 120^	ext{o}$$, what is the length of side $$c$$?

Accepted Answer

$$c = \sqrt{10^2$ + 7^2$ - 2 	imes 10 	imes 7 	imes \cos 120^	ext{o}}$$

Question 7

A triangle has sides $$a = 9$$, $$b = 12$$, and $$c = 15. $$Which formula would you use to find the angle opposite side $$c$$?

Accepted Answer

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

Question 8

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

Accepted Answer

$$C = \cos$$^{-1}$$ \left( \frac{6^2$ + 8^2$ - 10^2$}{2 	imes 6 	imes 8} ight)$$

Question 9

A triangle has sides $$a = 5$$, $$b = 7$$, and $$c = 9. $$Which of the following is the correct expression for finding angle $$C$$?

Accepted Answer

$$C = \cos$$^{-1}$$ \left( \frac{5^2$ + 7^2$ - 9^2$}{2 	imes 5 	imes 7} ight)$$

Question 10

A triangle has sides $$a = 13$$, $$b = 14$$, and $$c = 15. $$What is the value of $$\cos C$$?

Accepted Answer

$$\cos C = \frac{13^2$ + 14^2$ - 15^2$}{2 	imes 13 	imes 14}$$

Question 11

A triangle has sides $$a = 7$$, $$b = 24$$, and $$c = 25. $$What is the measure of angle $$C$$?

Accepted Answer

$$C = \cos$$^{-1}$$ \left( \frac{7^2$ + 24^2$ - 25^2$}{2 	imes 7 	imes 24} ight)$$

Question 12

A triangle has sides $$a = 8$$, $$b = 15$$, and $$c = 17. $$Which of the following is the correct formula for finding angle $$A$$?

Accepted Answer

$$A = \cos$$^{-1}$$ \left( \frac{15^2$ + 17^2$ - 8^2$}{2 	imes 15 	imes 17} ight)$$

The Law of Cosines: Applications and Problem Solving

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