Question 1

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

Accepted Answer

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

Question 2

A surveyor is standing at point $$A$$ and measures the angle between two landmarks $$B$$ and $$C as$$ $$50.5^\circ$$ and $$95.6^\circ$$ respectively. If the distance from $$A to$$ $$C is$$ $$47.5$$ meters, what is the distance from $$A to$$ $$B$$? (Use the Law of Sines.)

Accepted Answer

$$36.8$$ meters

Question 3

Given triangle $$ABC$$ with $$A = 40.0^\circ$$, $$a = 60.0 cm$$, and $$b = 35.0 cm$$, how many possible triangles can be constructed with these measurements (SSA case)?

Accepted Answer

One triangle

Question 4

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

Accepted Answer

Given two sides and an angle not included between them, the side opposite the given angle is shorter than the other given side.

Question 5

A pilot observes two landmarks due east. The landmarks are $$1540 m $$apart, and the angles of elevation from the pilot to the landmarks are $$32.0^\circ$$ and $$44.0^\circ. $$How far is the pilot from the closer landmark?

Accepted Answer

$$5145 m$$

Question 6

Which of the following is the correct Law of Cosines formula for finding side $$c in $$triangle $$ABC$$?

Accepted Answer

$$c^2$$ = $$a^2$$ + $$b^2 - 2ab$$ \cos C$$

Question 7

Given triangle $$ABC$$ with $$a = 25.5$$, $$b = 15.3$$, and $$c = 12.1$$, which law should you use first to find angle $$A$$?

Accepted Answer

Law of Cosines

Question 8

A nuclear submarine leaves its base and travels $$23.5 km/h $$for $$2.00$$ hours along a course of $$32.1^\circ$$ north of west. It then turns an additional $$21.5^\circ$$ north of west and travels for another $$1.00$$ hour. How far is the submarine from its base?

Accepted Answer

$$69.4 km$$

Question 9

Given triangle $$ABC$$ with $$a = 49.33 cm$$, $$b = 21.61 cm$$, $$c = 42.57 cm$$, what is the measure of angle $$A ($$opposite side $$a$$)?

Accepted Answer

$$94.81^\circ$$

Question 10

A motorist travels toward a mountain at $$75 km/h. $$The angle of elevation to the mountaintop changes from $$20^\circ to$$ $$30^\circ in$$ $$20$$ minutes. How far is the motorist from the mountaintop after $$20$$ minutes?

Accepted Answer

$$49.2 km$$

Question 11

Which of the following is NOT a valid case for applying the Law of Cosines?

Accepted Answer

Given two angles and a side (AAS or ASA)

Question 12

Given triangle $$ABC$$ with $$b = 30.0 cm$$, $$c = 75.0 cm$$, and $$A = 62.1^\circ$$, what is the correct sequence to solve for all sides and angles?

Accepted Answer

Use Law of Cosines to find $$a$$, then Law of Sines to find $$B$$, then angle sum to find $$C$$

Question 13

A plane leaves an airport and travels $$624 km $$due east, then turns less than $$90^\circ$$ toward the north and travels $$326 km$$, then turns again and travels $$846 km $$directly back to the airport. What is the measure of the first turn?

Accepted Answer

$$57.3^\circ$$

Question 14

If you are given $$a = 20.5$$, $$b = 10.3$$, and $$C = 73.3^\circ$$, what is the value of $$a^2$$ using the Law of Cosines?

Accepted Answer

$$a^2$$ = $$20.5^2$$ + $$10.3^2 - 2(20.5)(10.3$$) \cos 73.3^\circ$$

Question 15

Which of the following statements is TRUE regarding the Law of Sines and Law of Cosines?

Accepted Answer

The Law of Sines can be used to solve for an angle only if you know at least one side and its opposite angle.

Study Notes: Law of Sines and Law of Cosines for Oblique Triangles

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