Trigonometry problems on angle of elevation and depression.

Let’s solve each problem one by one, using trigonometry (specifically right triangle trig: sine, cosine, tangent).

---

Problem 5

> A man flies a kite with a 100 foot string. The angle of elevation of the string is 52°. How high off the ground is the kite?

Diagram Interpretation:
- The string = hypotenuse = 100 ft
- Angle of elevation = 52° (from ground to kite)
- Height of kite = opposite side to the angle

We use sine:

> sin(θ) = opposite / hypotenuse
> sin(52°) = height / 100
> height = 100 × sin(52°)

Use calculator:
> sin(52°) ≈ 0.7880
> height ≈ 100 × 0.7880 = 78.8 feet

✔ Answer: 78.8 feet

---

Problem 6

> From the top of a vertical cliff 40 m high, the angle of depression of an object that is level with the base of the cliff is 34°. How far is the object from the base of the cliff?

Important: Angle of depression = angle below horizontal. But in the right triangle formed, it’s equal to the angle of elevation from the object to the top of the cliff.

So:
- Opposite side = height of cliff = 40 m
- Angle (at object) = 34°
- We want adjacent side = distance from base of cliff to object

Use tangent:

> tan(θ) = opposite / adjacent
> tan(34°) = 40 / distance
> distance = 40 / tan(34°)

Calculate:
> tan(34°) ≈ 0.6745
> distance ≈ 40 / 0.6745 ≈ 59.3 meters

✔ Answer: 59.3 meters

---

Problem 7

> An airplane takes off 200 yards in front of a 60 foot building. At what angle of elevation must the plane take off in order to avoid crashing into the building? Assume that the airplane flies in a straight line and the angle of elevation remains constant until the airplane flies over the building.

Note: Units are mixed — 200 yards, 60 feet. Convert to same unit.

1 yard = 3 feet → 200 yards = 600 feet

Now we have:
- Opposite side = 60 ft (height of building)
- Adjacent side = 600 ft (horizontal distance to building)
- We want angle of elevation θ

Use tangent:

> tan(θ) = opposite / adjacent = 60 / 600 = 0.1
> θ = arctan(0.1)

Calculate:
> θ ≈ arctan(0.1) ≈ 5.7°

✔ Answer: 5.7°

---

Problem 8

> A 14 foot ladder is used to scale a 13 foot wall. At what angle of elevation must the ladder be situated in order to reach the top of the wall?

This forms a right triangle:
- Hypotenuse = ladder = 14 ft
- Opposite side = wall height = 13 ft
- We want angle θ between ground and ladder (angle of elevation)

Use sine:

> sin(θ) = opposite / hypotenuse = 13 / 14
> θ = arcsin(13/14)

Calculate:
> 13 ÷ 14 ≈ 0.9286
> θ ≈ arcsin(0.9286) ≈ 68.2°

✔ Answer: 68.2°

---

## ✔ Final Answers:

5. 78.8 feet
6. 59.3 meters
7. 5.7°
8. 68.2°

Let me know if you’d like diagrams or step-by-step calculator instructions!