Angles of Elevation Depression Worksheet | PDF | Teaching ... - Free Printable

Let’s solve each problem step by step using right triangle trigonometry. We’ll use the basic trigonometric ratios:
- SOH: sin(θ) = Opposite / Hypotenuse
- CAH: cos(θ) = Adjacent / Hypotenuse
- TOA: tan(θ) = Opposite / Adjacent

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Problem 1:

> The angle of elevation from point A to the top of a hill is 49°. If point A is 400 feet from the base of the hill, how high is the hill?

We are given:
- Angle of elevation = 49°
- Adjacent side (horizontal distance) = 400 ft
- We need to find the opposite side (height of the hill)

Use tan(θ) = opposite / adjacent

So:
> tan(49°) = height / 400

Multiply both sides by 400:
> height = 400 × tan(49°)

Using calculator:
> tan(49°) ≈ 1.1504
> height ≈ 400 × 1.1504 ≈ 460.16 ft

✔ Answer: 460 feet (rounded to nearest whole number)

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Problem 2:

> Find the angle of elevation of the sun when a 12.5-meter-tall telephone pole casts a 18-meter-long shadow.

We are given:
- Opposite side (height of pole) = 12.5 m
- Adjacent side (shadow length) = 18 m
- We need to find the angle θ

Use tan(θ) = opposite / adjacent

So:
> tan(θ) = 12.5 / 18 ≈ 0.6944

Now take arctan (inverse tangent):
> θ = arctan(0.6944) ≈ 34.78°

✔ Answer: 35° (rounded to nearest whole degree)

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Problem 3:

> A ladder leaning against a building makes an angle of 78° with the ground. The foot of the ladder is 5 feet from the building. How long is the ladder?

We are given:
- Angle = 78°
- Adjacent side (distance from wall) = 5 ft
- We need to find the hypotenuse (length of ladder)

Use cos(θ) = adjacent / hypotenuse

So:
> cos(78°) = 5 / ladder_length

Solve for ladder_length:
> ladder_length = 5 / cos(78°)

Using calculator:
> cos(78°) ≈ 0.2079
> ladder_length ≈ 5 / 0.2079 ≈ 24.05 ft

✔ Answer: 24 feet (rounded to nearest whole number)

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Problem 4:

> A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower. That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?

First, we must find the vertical distance from the person’s eyes to the window:

> Height difference = 132 ft – 5 ft = 127 ft

Horizontal distance = 100 ft (given)

We now have a right triangle where:
- Opposite = 127 ft
- Adjacent = 100 ft
- We need angle θ

Use tan(θ) = opposite / adjacent

> tan(θ) = 127 / 100 = 1.27

Take arctan:
> θ = arctan(1.27) ≈ 51.78°

✔ Answer: 52° (rounded to nearest whole degree)

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## ✔ Final Answers:

1. 460 feet
2. 35°
3. 24 feet
4. 52°

Make sure to show all your steps as required — especially writing the trig ratio used and rounding correctly. Great job practicing angles of elevation!