Question 1

A surveyor measures the angle of elevation to the top of a building from two points on level ground. From the first point, the angle is $$32.0^\circ$$ and from the second point, which is 1540 m closer to the building, the angle is $$44.0^\circ. $$How far is the first point from the top of the building?

Accepted Answer

$$5145.3\ 	ext{m}$$

Question 2

Given $$a = 20.3$$, $$b = 13.7$$, and $$C = 125.3^\circ$$, use the Law of Cosines to find $$c.$$

Accepted Answer

$$c = 30.4$$

Question 3

A triangle has sides $$a = 15.8\ 	ext{km}$$, $$b = 32.7\ 	ext{km}$$, and included angle $$C = 14.0^\circ. $$What is the length of side $$c$$?

Accepted Answer

$$c = 17.8\ 	ext{km}$$

Question 4

A motorist observes the angle of elevation to a mountain top change from $$20^\circ to$$ $$30^\circ$$ over a 20-minute period while driving at $$75\ 	ext{km/h}. $$How far is the motorist from the mountain after 20 minutes?

Accepted Answer

$$49.2\ 	ext{km}$$

Question 5

Given $$a = 49.33$$, $$b = 21.61$$, and $$c = 42.57$$, what is the largest angle of the triangle?

Accepted Answer

$$A = 94.81^\circ$$

Question 6

A submarine travels $$47.0\ 	ext{km} on a $$course of $$32.1^\circ$$ north of west, then turns $$21.5^\circ$$ further north and travels another $$23.5\ 	ext{km}. $$What is the total angle between its final and initial direction?

Accepted Answer

$$158.5^\circ$$

Question 7

Given $$a = 30.2$$, $$b = 12.4$$, $$c = 20.5$$, use the Law of Cosines to find angle $$A.$$

Accepted Answer

$$A = 131.1^\circ$$

Question 8

A triangle has $$A = 14.8^\circ$$, $$B = 121^\circ$$, and $$C = 44.2^\circ$$, with $$c = 729\ 	ext{cm}. $$Use the Law of Sines to find $$a.$$

Accepted Answer

$$a = 267.1\ 	ext{cm}$$

Question 9

Given $$a = 624\ 	ext{km}$$, $$b = 326\ 	ext{km}$$, and $$c = 846\ 	ext{km}$$, what is the measure of the largest angle in the triangle?

Accepted Answer

$$C = 122.7^\circ$$

Question 10

A triangle has $$a = 13.4$$, $$b = 5.63$$, and $$C = 42.3^\circ. $$Use the Law of Sines to find $$c.$$

Accepted Answer

$$c = 8.28$$

Question 11

Given $$a = 11.3$$, $$b = 28.2$$, and $$c = 18.2$$, use the Law of Cosines to find angle $$B.$$

Accepted Answer

$$B = 144.8^\circ$$

Question 12

A triangle has $$A = 31.1^\circ$$, $$b = 8.93$$, and $$c = 14.4. $$Use the Law of Sines to find $$a.$$

Accepted Answer

$$a = 8.93 \cdot \dfrac{\sin 31.1^\circ}{\sin 56.3^\circ} = 5.15$$

Question 13

A pilot flies 846 km, turns less than $$180^\circ$$ and travels 326 km, then turns again and travels 624 km back to the airport. What is the angle of the first turn?

Accepted Answer

$$57.3^\circ$$

Question 14

A triangle has $$a = 7.31$$, $$A = 74.4^\circ$$, and $$C = 25.3^\circ. $$Use the Law of Sines to find $$c.$$

Accepted Answer

$$c = 3.24$$

Question 15

A triangle has $$a = 13.4$$, $$A = 42.3^\circ$$, and $$C = 42.3^\circ. $$Use the Law of Sines to find $$c.$$

Accepted Answer

$$c = 13.4$$

Question 16

A triangle has $$A = 20^\circ$$, $$B = 150^\circ$$, and $$c = 25\ 	ext{km}. $$Use the Law of Sines to find $$a.$$

Accepted Answer

$$a = 49.2\ 	ext{km}$$

Law of Sines and Law of Cosines: Applications in Triangle Solving

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