Question 1

To solve a triangle means to find the __________ of its sides and all its __________.

Accepted Answer

lengths; angles

Question 2

If a triangle is NOT a right triangle, it is called a(n) __________ triangle.

Accepted Answer

oblique

Question 3

Which of the following is the Law of Sines for triangle $$ABC$$?

Accepted Answer

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

Question 4

Given triangle $$ABC$$ with $$A = 70^\circ$$, $$B = 80^\circ$$, and $$a = 12 m$$, what is the measure of angle $$C$$?

Accepted Answer

$$C = 180^\circ - 70^\circ - 80^\circ = 30^\circ$$

Question 5

In which cases can the Law of Sines be used to solve a triangle?

Accepted Answer

AAS, ASA, and SSA

Question 6

Given triangle $$ABC$$ with $$C = 115^\circ$$, $$B = 30^\circ$$, and $$a = 12 m$$, what is the measure of angle $$A$$?

Accepted Answer

$$A = 35^\circ$$

Question 7

Which of the following best describes the ambiguous case (SSA) when using the Law of Sines?

Accepted Answer

There may be zero, one, or two possible triangles.

Question 8

A palm tree leans toward an observer at an angle of $$78^\circ. At a $$distance of 60 feet from the base of the tree in the direction of the observer, the angle of elevation to the top of the tree is $$41^\circ. $$What is the length of the palm tree (to the nearest foot)?

Accepted Answer

About 67 feet

Question 9

Given triangle $$ABC$$ with $$A = 100^\circ$$, $$a = 50$$, and $$b = 34$$, which formula would you use to find angle $$B$$?

Accepted Answer

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

Question 10

If you are given two sides and an angle NOT between them (SSA), what must you check before solving the triangle?

Accepted Answer

The number of possible triangles (0, 1, or 2)

Question 11

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

Accepted Answer

When given all three sides (SSS)

Question 12

Given triangle $$ABC$$ with $$a = 10$$, $$A = 30^\circ$$, and $$b = 20$$, how many possible triangles can be formed?

Accepted Answer

Two

Question 13

If $$\frac{a}{\sin A} = \frac{b}{\sin B}$$, and $$a = 8$$, $$A = 45^\circ$$, $$b = 10$$, what is $$B$$?

Accepted Answer

$$B = \arcsin\left(\frac{10 \sin 45^\circ}{8}\right)$$

Question 14

Which of the following is a necessary condition for the ambiguous case to have two solutions?

Accepted Answer

$$a  b \sin A$$

Question 15

If $$A = 40^\circ$$, $$a = 12$$, and $$b = 20$$, what is the value of $$\sin B$$ using the Law of Sines?

Accepted Answer

$$\sin B = \frac{20 \sin 40^\circ}{12}$$

Question 16

A surveyor measures two angles of a triangular plot as $$55^\circ$$ and $$65^\circ$$, and the side between them as 100 meters. What is the length of the side opposite the $$55^\circ$$ angle?

Accepted Answer

$$\frac{100 \sin 55^\circ}{\sin 60^\circ}$$

Solving Oblique Triangles: The Law of Sines

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