Geometry worksheet with trigonometric ratio word problems and solutions.

Let’s solve each problem one by one using trigonometric ratios. We’ll use sine, cosine, or tangent depending on what sides and angles we’re given.

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Problem 1: How tall is the building?

We have a right triangle:
- Angle = 8°
- Adjacent side (ground) = 1000 m
- Opposite side = height of building (what we want)

Use tangent:
tan(θ) = opposite / adjacent
→ tan(8°) = height / 1000
→ height = 1000 × tan(8°)

Calculate tan(8°):
tan(8°) ≈ 0.1405
→ height ≈ 1000 × 0.1405 = 140.5 m

✔ Matches the answer given.

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Problem 2: How far up will the ladder reach?

Ladder = hypotenuse = 12 m
Angle with ground = 70°
We want the vertical height → that’s the opposite side.

Use sine:
sin(θ) = opposite / hypotenuse
→ sin(70°) = height / 12
→ height = 12 × sin(70°)

sin(70°) ≈ 0.9397
→ height ≈ 12 × 0.9397 ≈ 11.276 → round to nearest tenth = 11.3 m

✔ Matches the answer given.

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Problem 3: How tall is the monument?

We are given:
- Distance from base = 50 ft (adjacent)
- Angle of elevation = 85°
- Want height = opposite

Use tangent:
tan(85°) = height / 50
→ height = 50 × tan(85°)

tan(85°) ≈ 11.4301
→ height ≈ 50 × 11.4301 ≈ 571.505 → round to nearest tenth = 571.5 ft

✔ Matches the answer given.

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Problem 4: How high above the ground is the kite?

Kite string = hypotenuse = 155 m
Angle with ground = 73°
Want height = opposite

Use sine:
sin(73°) = height / 155
→ height = 155 × sin(73°)

sin(73°) ≈ 0.9563
→ height ≈ 155 × 0.9563 ≈ 148.2265 → round to nearest tenth = 148.2 m

✔ Matches the answer given.

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Problem 5: How far from the building is the car?

Building height = 25 m (opposite)
Angle of depression = 48° → this equals angle of elevation from car to top of building (alternate interior angles)

So in the right triangle:
- Opposite = 25 m
- Angle = 48°
- Adjacent = distance from building (what we want)

Use tangent:
tan(48°) = 25 / distance
→ distance = 25 / tan(48°)

tan(48°) ≈ 1.1106
→ distance ≈ 25 / 1.1106 ≈ 22.51 → round to nearest tenth = 22.5 m

✔ Matches the answer given.

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Problem 6: How wide is the river?

We have a right triangle:
- Height (vertical drop) = 15 m (opposite)
- Angle of depression = 10° → so angle at observer’s eye looking down is 10°, meaning angle inside triangle at top is also 10° (since horizontal line and riverbank are parallel)

Wait — actually, if you're standing 15 m above the river and look down at an angle of 10° to see the other bank, then:

The 15 m is the opposite side relative to the 10° angle.
The width of the river is the adjacent side.

So:
tan(10°) = opposite / adjacent = 15 / width
→ width = 15 / tan(10°)

tan(10°) ≈ 0.1763
→ width ≈ 15 / 0.1763 ≈ 85.08 → round to nearest tenth = 85.1 m

✔ Matches the answer given.

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Final Answer:
All answers match the provided solutions:
1. 140.5 m
2. 11.3 m
3. 571.5 ft
4. 148.2 m
5. 22.5 m
6. 85.1 m