Question 1

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

Accepted Answer

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

Question 2

In which of the following cases can the Law of Sines be used to solve an oblique triangle?

Accepted Answer

SAA, ASA, and SSA cases

Question 3

Given triangle $$ABC$$ with $$A = 40^\circ$$, $$B = 60^\circ$$, and $$a = 10$$ units, which Law of Sines proportion would you use to find side $$b$$?

Accepted Answer

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

Question 4

A surveyor measures two angles of a triangular plot as $$45^\circ$$ and $$70^\circ$$, and the side between them as 100 meters. Which case is this, and can the Law of Sines be used?

Accepted Answer

ASA case; Law of Sines can be used

Question 5

If $$\frac{a}{\sin A} \u003c b \u003c a$$, how many triangles can be formed in the SSA case?

Accepted Answer

Two oblique triangles

Question 6

A triangle has $$a = 8$$, $$A = 30^\circ$$, and $$b = 16. $$What is the value of $$\frac{a}{\sin A}$$, and what does it indicate about the number of possible triangles?

Accepted Answer

$$\frac{8}{\sin 30^\circ} = 16$$; one right triangle

Question 7

Which scenario describes the ambiguous case (SSA) for the Law of Sines?

Accepted Answer

Two sides and a non-included angle are known

Question 8

If $$a = 5$$, $$A = 40^\circ$$, and $$b = 8$$, what is the value of $$\frac{a}{\sin A}$$? Is it possible to form a triangle?

Accepted Answer

$$\frac{5}{\sin 40^\circ} \approx 7.78$$; yes, one oblique triangle

Question 9

A triangle has $$A = 50^\circ$$, $$B = 60^\circ$$, and $$a = 12. $$What is the first step to find side $$b$$ using the Law of Sines?

Accepted Answer

Set up $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ and solve for $$b$$

Question 10

In a triangle, if $$a = 7$$, $$A = 30^\circ$$, and $$b = 3$$, what is the value of $$\frac{a}{\sin A}$$ and how many triangles can be formed?

Accepted Answer

$$\frac{7}{\sin 30^\circ} = 14$$; no triangle

Question 11

A civil engineer needs to find the length of a bridge across a river. She measures the angles on both banks as $$45^\circ$$ and $$60^\circ$$, and the distance between the banks as 200 meters. Which Law of Sines proportion should she use?

Accepted Answer

$$\frac{200}{\sin 75^\circ} = \frac{x}{\sin 45^\circ}$$

Question 12

Which of the following is NOT a case where the Law of Sines can be used?

Accepted Answer

SAS (Side-Angle-Side)

Question 13

If $$a = 10$$, $$A = 45^\circ$$, and $$b = 7$$, what is the value of $$\frac{a}{\sin A}$$? Is $$b$$ long enough to form a triangle?

Accepted Answer

$$\frac{10}{\sin 45^\circ} \approx 14.14$$; $$b is $$too short, no triangle

Question 14

A triangle has $$A = 35^\circ$$, $$B = 65^\circ$$, and $$a = 15. $$What is the measure of angle $$C$$?

Accepted Answer

$$C = 80^\circ$$

Question 15

Which of the following best describes the process for solving an SAA triangle using the Law of Sines?

Accepted Answer

Find the unknown angle, then use Law of Sines to find the unknown sides

Question 16

A triangle has $$a = 9$$, $$A = 50^\circ$$, and $$b = 7. $$What is the value of $$\frac{a}{\sin A}$$ and how many triangles can be formed?

Accepted Answer

$$\frac{9}{\sin 50^\circ} \approx 11.76$$; $$b is $$too short, no triangle

Question 17

In the SSA case, if $$b = a \sin A$$, what type of triangle is formed?

Accepted Answer

One right triangle

Question 18

A triangle has $$A = 40^\circ$$, $$B = 60^\circ$$, and $$a = 10. $$What is the value of $$\frac{a}{\sin A}$$?

Accepted Answer

$$\frac{10}{\sin 40^\circ} \approx 15.57$$

The Law of Sines and Its Applications in Solving Oblique Triangles

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