Q: Which of the following is true about the Law of Sines and supplementary angles?

$$\sin \theta = \sin (180^\circ - \theta)$$

Question 1

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

Accepted Answer

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

Question 2

A surveyor wants to find the width of a river. From one bank, she measures an angle of $$31.1^\circ to a $$point directly across the river and an angle of $$112.9^\circ to a $$point downstream. The distance between these two points on her side is $$347.6 ft. $$What is the width of the river?

Accepted Answer

$$305.5 ft$$

Question 3

Which of the following sets of information is NOT sufficient to solve an oblique triangle uniquely?

Accepted Answer

Three angles (AAA)

Question 4

Given triangle $$ABC$$ with $$A = 43.5^\circ$$, $$a = 10.7 in$$, and $$c = 7.2 in$$, what is the measure of angle $$C$$?

Accepted Answer

$$27.6^\circ$$

Question 5

Which scenario describes the ambiguous case (SSA) when solving triangles using the Law of Sines?

Accepted Answer

Two sides and an angle opposite one of them are known

Question 6

If $$A = 104^\circ$$, $$a = 26.8 m$$, and $$b = 31.3 m in $$triangle $$ABC$$, why is this not a valid triangle?

Accepted Answer

Because $$A is $$obtuse and $$a$$ must be the longest side, but $$b \u003e a$$

Question 7

What is the area of a triangle with sides $$a = 34.0 ft$$, $$c = 42.0 ft$$, and included angle $$B = 55^\circ 10'$$?

Accepted Answer

$$A = \dfrac{1}{2}ac\sin B = \dfrac{1}{2}(34.0)(42.0)\sin 55^\circ 10' \approx 586 ft$^2$$

Question 8

In the ambiguous case (SSA), if $$a = 22.8 ft$$, $$b = 24.9 ft$$, and $$A = 55.3^\circ$$, how many triangles can be formed?

Accepted Answer

Two triangles

Question 9

Which of the following is a correct formula for the area of a triangle given two sides and the included angle?

Accepted Answer

$$A = \dfrac{1}{2}ab\sin C$$

Question 10

A forest fire is located on a bearing N $$42^\circ E $$from station $$A$$ and N $$15^\circ E $$from station $$B$$, which are $$110 mi $$apart on an east-west line. What is the distance from station $$A to $$the fire (to the nearest ten miles)?

Accepted Answer

$$230.5 mi$$

Question 11

If $$\sin B \u003e 1$$ when applying the Law of Sines in the SSA case, what does this indicate?

Accepted Answer

No triangle satisfies the given conditions

Question 12

Given $$a = 42.9 cm$$, $$A = 32.0^\circ$$, and $$C = 81.8^\circ$$, what is the value of $$b in $$triangle $$ABC$$?

Accepted Answer

$$b = \dfrac{42.9 \sin 81.8^\circ}{\sin 32.0^\circ} \approx 80.1 cm$$

Question 13

Which congruence axiom states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent?

Accepted Answer

Side-Angle-Side (SAS)

Question 14

If $$a = 10.7 in$$, $$A = 43.5^\circ$$, and $$C = 27.6^\circ$$, what is the value of $$b in $$triangle $$ABC$$?

Accepted Answer

$$b = \dfrac{10.7 \sin 108.9^\circ}{\sin 43.5^\circ} \approx 14.7 in$$

Question 15

Which of the following is true about the Law of Sines and supplementary angles?

Accepted Answer

$$\sin 	heta = \sin (180^\circ - 	heta)$$

Chapter 8: Applications of Trigonometry – The Law of Sines

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