Question 1

Given a triangle with sides $a = 7$, $b = 9$, and angle $C = 35^	ext{o}$, 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 $c$ for a triangle with $a = 7$, $b = 9$, and $C = 35^	ext{o}$. Which of the following is closest to the correct answer?

Accepted Answer

$c \approx 5.7$

Question 3

Which of the following is the correct Law of Sines equation for a triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$?

Accepted Answer

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Question 4

A surveyor measures two sides of a triangular plot of land as $a = 120$ m and $b = 150$ m, with the angle between them $C = 60^	ext{o}$. What is the area of the plot?

Accepted Answer

$A = 7794.23$ m$^2$

Question 5

If $a = 8$, $b = 15$, and $C = 45^	ext{o}$, what is the value of $c$ using the Law of Cosines?

Accepted Answer

$c = \sqrt{8^2 + 15^2 - 2 \cdot 8 \cdot 15 \cdot \cos 45^	ext{o}}$

Question 6

Which scenario best describes when to use the Law of Sines instead of the Law of Cosines?

Accepted Answer

When you know two angles and one side.

Question 7

A triangle has $A = 40^	ext{o}$, $B = 60^	ext{o}$, and $a = 10$. What is the length of side $b$?

Accepted Answer

$b = 10 \cdot \frac{\sin 60^	ext{o}}{\sin 40^	ext{o}}$

Question 8

A pilot needs to calculate the distance between two cities using triangulation. If the angle at the airport is $50^	ext{o}$ and the distances to the cities are $a = 200$ km and $b = 250$ km, what formula should be used to find the distance between the cities?

Accepted Answer

Law of Cosines

Question 9

Which of the following is NOT a valid application of the Law of Sines?

Accepted Answer

Finding the area of a triangle when all three sides are known.

Question 10

A triangle has sides $a = 5$, $b = 7$, and $c = 10$. Which formula would you use to find angle $C$?

Accepted Answer

Law of Cosines

Question 11

A construction engineer needs to find the height of a triangular support beam. If the base is $12$ m, the side is $15$ m, and the angle between them is $30^	ext{o}$, what is the height?

Accepted Answer

$h = 15 \cdot \sin 30^	ext{o}$

Question 12

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 13

A triangle has $a = 6$, $b = 8$, and $C = 90^	ext{o}$. What is the length of side $c$?

Accepted Answer

$c = \sqrt{6^2 + 8^2}$

Question 14

Which of the following is a real-world application of the Law of Sines?

Accepted Answer

Calculating the height of a mountain using angles and distances measured from two points.

The Law of Sines in Trigonometry

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