Q: What is the angle between the diagonal of a cube and the diagonal of its base? (Let the side of the cube be $$a.)$$

$$\theta = \arccos\left(\frac{1}{\sqrt{3}}\right) \approx 54.7^\circ$$

Q: Find the angle of inclination of the line $$y = \frac{2}{3}x + 12.$$

$$\theta = \arctan\left(\frac{2}{3}\right) \approx 33.7^\circ$$

Q: Find the angle of inclination of the line $$y = 3x - 7.$$

$$\theta = \arctan(3) \approx 71.6^\circ$$

Q: A surveyor stands 300 meters from the base of a building and measures the angle of elevation to the top as $$45^\circ. $$What is the height of the building?

$$h = 300 \tan 45^\circ = 300 m$$

Question 1

A camera is placed 300 meters away from the launch point of a hot air balloon. If the balloon rises vertically to a height $$x$$, what is the correct formula for the angle of elevation $$t (in $$radians) from the camera to the balloon?

Accepted Answer

$$t = \arctan\left(\frac{x}{300}\right)$$

Question 2

If the hot air balloon in the previous problem is at a height of $$x = 150$$ meters, what is the angle of elevation $$t (in $$degrees) from the camera to the balloon? (Round to 1 decimal place.)

Accepted Answer

$$t \approx 26.6^\circ$$

Question 3

A lighthouse is 350 feet above sea level. The angles of depression from the lighthouse to two ships are $$40^\circ$$ and $$6.5^\circ. $$What is the correct formula to find the distance $$d$$ between the two ships?

Accepted Answer

$$d = 350 \left( \cot 6.5^\circ - \cot 40^\circ \right)$$

Question 4

A passenger in an airplane at an altitude of 10 km sees two towns directly east. The angles of depression to the towns are $$28^\circ$$ and $$55^\circ. $$What is the distance between the two towns? (Round to 2 decimal places.)

Accepted Answer

$$d = 10 \left( \cot 28^\circ - \cot 55^\circ \right) \approx 13.13 km$$

Question 5

What is the angle between the diagonal of a cube and the diagonal of its base? (Let the side of the cube be $$a.)$$

Accepted Answer

$$	heta = \arccos\left(\frac{1}{\sqrt{3}}ight) \approx 54.7^\circ$$

Question 6

Find the angle of inclination of the line $$y = \frac{2}{3}x + 12.$$

Accepted Answer

$$	heta = \arctan\left(\frac{2}{3}ight) \approx 33.7^\circ$$

Question 7

Find the angle of inclination of the line $$y = 3x - 7.$$

Accepted Answer

$$	heta = \arctan(3) \approx 71.6^\circ$$

Question 8

A surveyor stands 300 meters from the base of a building and measures the angle of elevation to the top as $$45^\circ. $$What is the height of the building?

Accepted Answer

$$h = 300 	an 45^\circ = 300 m$$

Question 9

A drone flies directly above a straight road. From a point on the road, the angle of elevation to the drone is $$30^\circ. If $$the drone is 200 meters above the ground, how far is the drone horizontally from the observer?

Accepted Answer

$$d = 200 \cot 30^\circ \approx 346.4 m$$

Question 10

A fire lookout tower is 50 meters tall. The angle of depression from the top of the tower to a fire on the ground is $$12^\circ. $$How far is the fire from the base of the tower?

Accepted Answer

$$d = 50 \cot 12^\circ \approx 236.1 m$$

Question 11

A 20-meter ladder leans against a wall, making an angle of $$60^\circ$$ with the ground. How high up the wall does the ladder reach?

Accepted Answer

$$h = 20 \sin 60^\circ \approx 17.3 m$$

Question 12

A tree casts a shadow 25 meters long when the angle of elevation of the sun is $$30^\circ. $$What is the height of the tree?

Accepted Answer

$$h = 25 	an 60^\circ \approx 43.3 m$$

Question 13

A building and a tree are 50 meters apart. From a point between them, the angles of elevation to the top of the building and the tree are $$40^\circ$$ and $$30^\circ$$ respectively. If the point is 20 meters from the building, how tall is the building?

Accepted Answer

$$h = 20 	an 40^\circ \approx 16.8 m$$

Question 14

A ramp rises 1 meter for every 4 meters of horizontal distance. What is the angle of inclination of the ramp?

Accepted Answer

$$	heta = \arctan\left(\frac{1}{4}ight) \approx 14.0^\circ$$

Trigonometry Application Problems: Step-by-Step Guidance

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