Q: A house has a roof with a pitch of $$\frac{10}{12}. $$What is the angle $$\theta$$ that the rafter makes with the base of the roof?

$$\theta = \arctan \left( \frac{10}{12} \right)$$

Q: A gondola travels $$5750 ft $$from a town at elevation $$8725 ft to a $$station at elevation $$10,550 ft. $$What is the angle of elevation from the town to the station?

$$\theta = \arctan \left( \frac{1825}{5750} \right)$$

Q: If $$\sin \theta = 0.6$$ and $$\theta is an $$acute angle in a right triangle, what is $$\cos \theta$$?

$$\cos \theta = \sqrt{1 - 0.6^2$}$$

Question 1

A right triangle has a hypotenuse of length 106.2 and one angle measuring $$61.7^\circ. $$What is the length of the side adjacent to the $$61.7^\circ$$ angle?

Accepted Answer

$$b = 106.2 \cos 61.7^\circ$$

Question 2

A house has a roof with a pitch of $$\frac{10}{12}. $$What is the angle $$	heta$$ that the rafter makes with the base of the roof?

Accepted Answer

$$	heta = \arctan \left( \frac{10}{12} ight)$$

Question 3

A rafter in a house forms a right triangle with the base of $$21 ft $$and an angle of $$39.8^\circ. $$What is the length $$x of $$the rafter?

Accepted Answer

$$x = \frac{21}{\cos 39.8^\circ}$$

Question 4

A pilot sees an airport at an angle of elevation of $$15^\circ$$ from a horizontal distance of $$2$$ miles. What is the altitude of the plane above the airport?

Accepted Answer

$$h = 2 	an 15^\circ$$

Question 5

A gondola travels $$5750 ft $$from a town at elevation $$8725 ft to a $$station at elevation $$10,550 ft. $$What is the angle of elevation from the town to the station?

Accepted Answer

$$	heta = \arctan \left( \frac{1825}{5750} ight)$$

Question 6

A ranger at point A sees a fire directly south. Another ranger at point B, $$7.5$$ miles east, sights the fire at a bearing of $$S 27.38^\circ W. $$How far is the fire from point A?

Accepted Answer

$$d = 7.5 	an 62.62^\circ$$

Question 7

From the last row of the upper deck at a baseball stadium, the angle of depression to home plate is $$29.9^\circ$$ and the vertical height is $$144 ft. $$What is the viewing distance to home plate?

Accepted Answer

$$d = \frac{144}{\sin 29.9^\circ}$$

Question 8

Which of the following best describes an angle of depression?

Accepted Answer

The angle between the horizontal and a line of sight below the horizontal

Question 9

A bearing of $$N 55^\circ W$$ means:

Accepted Answer

55 degrees west of north

Question 10

A gondola descends from Station St. Sophia (elevation $$10,550 ft) to $$Mountain Village (elevation $$9500 ft) $$over a distance of $$3913 ft. $$What is the angle of depression?

Accepted Answer

$$\beta = \arctan \left( \frac{1050}{3913} \right)$$

Question 11

A ladder leans against a wall making an angle of $$65^\circ$$ with the ground. If the foot of the ladder is $$8 ft $$from the wall, how high up the wall does the ladder reach?

Accepted Answer

$$h = 8 	an 65^\circ$$

Question 12

If $$\sin 	heta = 0.6$$ and $$	heta is an $$acute angle in a right triangle, what is $$\cos 	heta$$?

Accepted Answer

$$\cos 	heta = \sqrt{1 - 0.6^2$}$$

Question 13

A surveyor measures the angle of elevation to the top of a building as $$40^\circ$$ from a point $$50$$ meters from the base. What is the height of the building?

Accepted Answer

$$h = 50 	an 40^\circ$$

Question 14

A right triangle has legs $$a$$ and $$b$$, and hypotenuse $$c. If$$ $$a = 9$$ and $$c = 15$$, what is $$b$$?

Accepted Answer

$$b = \sqrt{15^2$ - 9^2$}$$

Question 15

A building casts a shadow $$30$$ meters long when the angle of elevation of the sun is $$35^\circ. $$How tall is the building?

Accepted Answer

$$h = 30 	an 35^\circ$$

Question 16

A ship sails on a bearing of $$S 67^\circ E. $$Which direction is the ship heading?

Accepted Answer

67 degrees east of south

Question 17

A right triangle has one angle of $$30^\circ$$ and a hypotenuse of $$20$$ units. What is the length of the side opposite the $$30^\circ$$ angle?

Accepted Answer

$$10$$

Applications of Right Triangles and Trigonometric Functions

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