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 $\cos C$?

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

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

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

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

Study Guide - Practice Questions

Study Guide - Flashcards