Worksheet featuring seven word problems that apply right triangle trigonometry to real-life scenarios like hiking, ladders, and kite flying.

Let’s solve each problem one by one, step by step. We’ll use right triangle trigonometry — that means we’ll use sine, cosine, or tangent depending on what sides and angles we’re given.

Remember:
- Tangent (tan) = opposite / adjacent → good when you have angle + two legs
- Sine (sin) = opposite / hypotenuse → good when you have angle + opposite side + hypotenuse
- Cosine (cos) = adjacent / hypotenuse → good when you have angle + adjacent side + hypotenuse

We’ll also round answers as instructed.

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Problem 1:
From the top of a lighthouse 100 feet above sea level, the angle of depression to a boat at sea is 25 degrees. What is the horizontal distance from the boat to the base of the lighthouse?

→ Angle of depression = angle below horizontal. So inside the triangle, it’s also 25° (alternate interior angles).

We have:
- Opposite side = height of lighthouse = 100 ft
- Adjacent side = horizontal distance (what we want)
- Angle = 25°

Use tan(θ) = opposite / adjacent
→ tan(25°) = 100 / x
→ x = 100 / tan(25°)

Calculate tan(25°): ≈ 0.4663
→ x = 100 / 0.4663 ≈ 214.45

Round to nearest foot → 214 feet

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Problem 2:
A ski slope on a mountain has an angle of elevation of 25.2 degrees. The vertical height of the slope is 1808 feet. How long is the ski slope?

→ “Angle of elevation” means looking up from bottom to top.

We have:
- Opposite side = vertical height = 1808 ft
- Hypotenuse = length of slope (what we want)
- Angle = 25.2°

Use sin(θ) = opposite / hypotenuse
→ sin(25.2°) = 1808 / x
→ x = 1808 / sin(25.2°)

Calculate sin(25.2°): ≈ 0.4258
→ x = 1808 / 0.4258 ≈ 4246.12

Round to nearest foot → 4246 feet

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Problem 3:
A man on a 130-foot cliff sees his friend at an angle of 16 degrees and sees his bicycle. To the nearest foot, how far away is the man from his bike if his bike lies from the base of the cliff?

Wait — this wording is confusing. Let me re-read:

“A man on a 130-foot vertical cliff looks down at an angle of 16 degrees and sees his bicycle. To the nearest foot, how far away is the man from his bike if his bike lies from the base of the cliff?”

I think it means: He looks down at 16° angle of depression to see his bike, which is on the ground at the base of the cliff? But then “how far away is the man from his bike” — that would be the straight-line distance (hypotenuse), not horizontal.

But let’s check: If he’s on a 130-ft cliff, and looks down at 16° to see the bike, and the bike is at the base of the cliff — that doesn’t make sense because if the bike is at the base, the angle of depression would be 90°.

Ah — probably typo or misphrasing. Likely: He looks down at 16° angle of depression to see his bike, which is some distance away from the base of the cliff. Then “how far away is the man from his bike” — meaning straight line distance (hypotenuse).

So:

- Vertical leg = 130 ft (opposite to angle)
- Angle of depression = 16° → so in triangle, angle at top is 16°
- We want hypotenuse (distance from man to bike)

Use sin(θ) = opposite / hypotenuse
→ sin(16°) = 130 / x
→ x = 130 / sin(16°)

sin(16°) ≈ 0.2756
→ x = 130 / 0.2756 ≈ 471.7

Round to nearest foot → 472 feet

*(Note: If they meant horizontal distance, it would be adjacent side, using tan. But question says “how far away is the man from his bike” — that’s direct distance, so hypotenuse.)*

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Problem 4:
A ladder leans against a wall. The top of the ladder reaches 15 feet up the wall, and makes an angle of 53 degrees with the wall. To the nearest tenth of a foot, how long is the ladder?

Important: Angle is with the wall, not the ground.

So if ladder makes 53° with the wall, then angle with ground is 90° - 53° = 37°.

But let’s draw it:

Wall is vertical. Ladder touches wall at 15 ft high. Angle between ladder and wall is 53°.

So in the right triangle:
- Side adjacent to 53° angle = 15 ft (along the wall)
- Hypotenuse = ladder length (what we want)

So cos(53°) = adjacent / hypotenuse = 15 / x
→ x = 15 / cos(53°)

cos(53°) ≈ 0.6018
→ x = 15 / 0.6018 ≈ 24.925

Round to nearest tenth → 24.9 feet

*(Alternatively, if you used angle with ground: 37°, then sin(37°) = 15/x → same result since sin(37)=cos(53))*

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Problem 5:
The angle from the base to the top of a water slide is 13 degrees. The slide extends horizontally for 58.2 meters. To the nearest tenth of a meter, what is the height of the water slide?

So:
- Horizontal distance (adjacent) = 58.2 m
- Angle at base = 13°
- Height = opposite side (what we want)

Use tan(θ) = opposite / adjacent
→ tan(13°) = h / 58.2
→ h = 58.2 * tan(13°)

tan(13°) ≈ 0.2309
→ h = 58.2 * 0.2309 ≈ 13.438

Round to nearest tenth → 13.4 meters

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Problem 6:
A person flying a kite has released 176 meters of string. The string makes an angle of 27 degrees with the ground. To the nearest meter, how high is the kite and, horizontally, how far away is the kite?

String = hypotenuse = 176 m
Angle with ground = 27°

Height = opposite side → use sin
Horizontal distance = adjacent side → use cos

Height:
h = 176 * sin(27°)
sin(27°) ≈ 0.4540
→ h = 176 * 0.4540 ≈ 79.904 → 80 meters

Horizontal distance:
d = 176 * cos(27°)
cos(27°) ≈ 0.8910
→ d = 176 * 0.8910 ≈ 156.816 → 157 meters

Answer: Height = 80 m, Horizontal distance = 157 m

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Problem 7:
An airplane is flying at an altitude of 6000 meters over the ocean directly toward a coastline. At a certain moment, the angle of depression from the plane to the coast is 14 degrees. To the nearest kilometer, how much farther does the plane have to fly until it is directly above the coastline?

Altitude = 6000 m = 6 km (since answer needs to be in km)

Angle of depression = 14° → so in triangle, angle at plane is 14°, opposite side = 6 km, adjacent side = horizontal distance to coast (what we want)

Use tan(θ) = opposite / adjacent
→ tan(14°) = 6 / x
→ x = 6 / tan(14°)

tan(14°) ≈ 0.2493
→ x = 6 / 0.2493 ≈ 24.067 km

Round to nearest kilometer → 24 kilometers

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Now, compiling all final answers:

Final Answer:
1. 214 feet
2. 4246 feet
3. 472 feet
4. 24.9 feet
5. 13.4 meters
6. Height: 80 meters, Horizontal distance: 157 meters
7. 24 kilometers