Q: A right triangle has an angle $$A = 34^ ext{\circ} 30'$$ and hypotenuse $$c = 12.7$$ inches. What is the length of side $$a$$ opposite angle $$A$$? (Round to two decimal places.)

$$a = 12.7 \sin 34^\text{\circ} 30' \approx 7.19$$ inches

Q: Given a right triangle with $$a = 29.43 cm $$and $$c = 53.58 cm$$, what is the measure of angle $$A ($$opposite $$a$$)?

$$A = \arcsin \left( \frac{29.43}{53.58} \right) \approx 33.32^\text{\circ}$$

Q: A building casts a shadow of 37.62 meters when the building is 34.09 meters tall. What is the angle of elevation of the sun?

$$\tan B = \frac{34.09}{37.62}$$, $$B = \arctan \left( \frac{34.09}{37.62} \right) \approx 42.18^\text{\circ}$$

Q: A radar station A is west of station B by 3.70 km. A plane is detected at C, with bearings of $$61.0^\text{\circ}$$ from A and $$331.0^\text{\circ}$$ from B. What is the distance from A to C?

$$b = 3.70 \cos 29.0^\text{\circ} \approx 3.24 km$$

Q: Given the bearings S $$52^\text{\circ} E $$from A to C, N $$84^\text{\circ} E $$from A to B, and S $$38^\text{\circ} W $$from B to C, and the distance from A to B is 600 miles, what is the distance from A to C if $$\angle BAC = 46^\text{\circ}$$?

$$b = 600 \sin 46^\text{\circ} \approx 430$$ miles

Q: If a runway is labeled 12L, what is the number of the corresponding right runway (R) and what is the bearing of the right runway?

30R, bearing $$300^\text{\circ}$$

Q: The parallax angle to Alpha Centauri is $$0.000212^\text{\circ}$$ and the Earth-Sun distance is 93,000,000 miles. What is the distance to Alpha Centauri in miles?

$$d = \frac{93,000,000}{\sin 0.000212^\text{\circ}} \approx 2.51 \times 10$$^{13}$$ miles

Q: If the distance to Alpha Centauri is $$2.51 \times 10$$^{13}$$ miles and a light-year is 5.9$$ \times $$10^{12}$$ miles, what is the distance to Alpha Centauri in light-years?

$$\frac{2.51 \times 10$$^{13}$$}{5.9 \times 10$$^{12}$$} \approx 4.3$$ light-years

Q: Francisco stands at a point where the angle of elevation to the top of a tree is $$36.7^\text{\circ}. He $$moves back 50 feet and the angle of elevation becomes $$22.2^\text{\circ}. $$What is the height of the tree?

$$h = \frac{50 \tan 22.2^\text{\circ}}{\tan 36.7^\text{\circ} - \tan 22.2^\text{\circ}} \tan 36.7^\text{\circ} \approx 45$$ feet

Question 1

A right triangle has an angle $$A = 34^	ext{\circ} 30'$$ and hypotenuse $$c = 12.7$$ inches. What is the length of side $$a$$ opposite angle $$A$$? (Round to two decimal places.)

Accepted Answer

$$a = 12.7 \sin 34^	ext{\circ} 30' \approx 7.19$$ inches

Question 2

Given a right triangle with $$a = 29.43 cm $$and $$c = 53.58 cm$$, what is the measure of angle $$A ($$opposite $$a$$)?

Accepted Answer

$$A = \arcsin \left( \frac{29.43}{53.58} ight) \approx 33.32^	ext{\circ}$$

Question 3

A building casts a shadow of 37.62 meters when the building is 34.09 meters tall. What is the angle of elevation of the sun?

Accepted Answer

$$	an B = \frac{34.09}{37.62}$$, $$B = \arctan \left( \frac{34.09}{37.62} ight) \approx 42.18^	ext{\circ}$$

Question 4

Which of the following best describes the angle of depression?

Accepted Answer

The acute angle formed by a line of sight downward from a horizontal line to an object below.

Question 5

A radar station A is west of station B by 3.70 km. A plane is detected at C, with bearings of $$61.0^	ext{\circ}$$ from A and $$331.0^	ext{\circ}$$ from B. What is the distance from A to C?

Accepted Answer

$$b = 3.70 \cos 29.0^	ext{\circ} \approx 3.24 km$$

Question 6

A plane flies from A to B on a bearing of N $$84^	ext{\circ} E $$for 2.4 hours at 250 mph. What is the distance from A to B?

Accepted Answer

$$250 	imes 2.4 = 600$$ miles

Question 7

Given the bearings S $$52^	ext{\circ} E $$from A to C, N $$84^	ext{\circ} E $$from A to B, and S $$38^	ext{\circ} W $$from B to C, and the distance from A to B is 600 miles, what is the distance from A to C if $$\angle BAC = 46^	ext{\circ}$$?

Accepted Answer

$$b = 600 \sin 46^	ext{\circ} \approx 430$$ miles

Question 8

If a runway is labeled 12L, what is the number of the corresponding right runway (R) and what is the bearing of the right runway?

Accepted Answer

30R, bearing $$300^	ext{\circ}$$

Question 9

The parallax angle to Alpha Centauri is $$0.000212^	ext{\circ}$$ and the Earth-Sun distance is 93,000,000 miles. What is the distance to Alpha Centauri in miles?

Accepted Answer

$$d = \frac{93,000,000}{\sin 0.000212^	ext{\circ}} \approx 2.51 	imes 10$$^{13}$$ miles

Question 10

If the distance to Alpha Centauri is $$2.51 	imes 10$$^{13}$$ miles and a light-year is 5.9$$ 	imes $$10^{12}$$ miles, what is the distance to Alpha Centauri in light-years?

Accepted Answer

$$\frac{2.51 	imes 10$$^{13}$$}{5.9 	imes 10$$^{12}$$} \approx 4.3$$ light-years

Question 11

Francisco stands at a point where the angle of elevation to the top of a tree is $$36.7^	ext{\circ}. He $$moves back 50 feet and the angle of elevation becomes $$22.2^	ext{\circ}. $$What is the height of the tree?

Accepted Answer

$$h = \frac{50 	an 22.2^	ext{\circ}}{	an 36.7^	ext{\circ} - 	an 22.2^	ext{\circ}} 	an 36.7^	ext{\circ} \approx 45$$ feet

Question 12

Which of the following is the correct process for solving an applied trigonometry problem involving right triangles?

Accepted Answer

Draw a sketch, write an equation relating the given quantities, solve the equation, and check your answer.

Question 13

If a right triangle has legs $$a = 29.43 cm $$and $$b = 44.77 cm$$, what is the length of the hypotenuse $$c$$?

Accepted Answer

$$c = \sqrt{29.43^2$ + 44.77^2$} \approx 53.58 cm$$

Question 14

Which of the following best describes the first method for expressing bearing?

Accepted Answer

A single angle measured clockwise from north.

Question 15

A surveyor needs to find the height of a building. From a point 80 meters from the base, the angle of elevation to the top is $$28^	ext{\circ}. $$What is the height of the building?

Accepted Answer

$$h = 80 	an 28^	ext{\circ} \approx 42.6$$ meters

Applications of Right Triangles in Trigonometry

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