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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