Trigonometry word problems worksheet with illustrated scenarios and mathematical equations.

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The image contains a worksheet titled "GEOMETRY - TRIGONOMETRY: ANGLE OF ELEVATION PROBLEMS" with 10 problems. Each problem involves finding an unknown side length or angle in a right triangle using trigonometric ratios (sine, cosine, tangent).

I will solve each problem one by one, showing the setup and calculation for each.

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Problem 1:
A 20 ft ladder leans against a wall. The base of the ladder is 8 ft from the wall. How high up the wall does the ladder reach?
- This is a right triangle. We know the hypotenuse (ladder = 20 ft) and the adjacent side to the angle at the ground (base = 8 ft). We need to find the opposite side (height).
- Use Pythagorean theorem: $a^2 + b^2 = c^2$
- Let $h$ be the height. $h^2 + 8^2 = 20^2$
- $h^2 + 64 = 400$
- $h^2 = 336$
- $h = \sqrt{336} \approx 18.33$ ft

Problem 2:
A 15 ft ladder leans against a wall. The top of the ladder reaches 12 ft up the wall. What is the angle the ladder makes with the ground?
- We know the opposite side (12 ft) and the hypotenuse (15 ft). We need to find the angle $\theta$.
- Use sine: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{15} = 0.8$
- $\theta = \arcsin(0.8)$
- $\theta \approx 53.13^\circ$

Problem 3:
From a point 50 ft from the base of a flagpole, the angle of elevation to the top of the pole is 60°. Find the height of the flagpole.
- We know the adjacent side (50 ft) and the angle (60°). We need to find the opposite side (height).
- Use tangent: $\tan(60^\circ) = \frac{\text{height}}{50}$
- $\text{height} = 50 \times \tan(60^\circ)$
- $\tan(60^\circ) = \sqrt{3} \approx 1.732$
- $\text{height} = 50 \times 1.732 = 86.6$ ft

Problem 4:
A kite is flying at a height of 60 ft. The string makes an angle of 45° with the ground. How long is the string?
- We know the opposite side (60 ft) and the angle (45°). We need to find the hypotenuse (string length).
- Use sine: $\sin(45^\circ) = \frac{60}{\text{string}}$
- $\text{string} = \frac{60}{\sin(45^\circ)}$
- $\sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.7071$
- $\text{string} = \frac{60}{0.7071} \approx 84.85$ ft

Problem 5:
A tree casts a shadow 20 ft long. At the same time, the angle of elevation of the sun is 30°. How tall is the tree?
- We know the adjacent side (shadow = 20 ft) and the angle (30°). We need to find the opposite side (tree height).
- Use tangent: $\tan(30^\circ) = \frac{\text{height}}{20}$
- $\text{height} = 20 \times \tan(30^\circ)$
- $\tan(30^\circ) = \frac{1}{\sqrt{3}} \approx 0.5774$
- $\text{height} = 20 \times 0.5774 \approx 11.55$ ft

Problem 6:
A 10 ft ladder leans against a wall. The angle between the ladder and the ground is 70°. How far is the base of the ladder from the wall?
- We know the hypotenuse (10 ft) and the angle (70°). We need to find the adjacent side (distance from wall).
- Use cosine: $\cos(70^\circ) = \frac{\text{distance}}{10}$
- $\text{distance} = 10 \times \cos(70^\circ)$
- $\cos(70^\circ) \approx 0.3420$
- $\text{distance} = 10 \times 0.3420 = 3.42$ ft

Problem 7:
From a point 30 ft from the base of a building, the angle of elevation to the top is 50°. Find the height of the building.
- We know the adjacent side (30 ft) and the angle (50°). We need to find the opposite side (height).
- Use tangent: $\tan(50^\circ) = \frac{\text{height}}{30}$
- $\text{height} = 30 \times \tan(50^\circ)$
- $\tan(50^\circ) \approx 1.1918$
- $\text{height} = 30 \times 1.1918 \approx 35.75$ ft

Problem 8:
A 12 ft ladder leans against a wall. The base of the ladder is 5 ft from the wall. What is the angle the ladder makes with the ground?
- We know the adjacent side (5 ft) and the hypotenuse (12 ft). We need to find the angle $\theta$.
- Use cosine: $\cos(\theta) = \frac{5}{12}$
- $\theta = \arccos(\frac{5}{12})$
- $\frac{5}{12} \approx 0.4167$
- $\theta \approx 65.38^\circ$

Problem 9:
A person stands 40 ft from a tower. The angle of elevation to the top of the tower is 35°. Find the height of the tower.
- We know the adjacent side (40 ft) and the angle (35°). We need to find the opposite side (height).
- Use tangent: $\tan(35^\circ) = \frac{\text{height}}{40}$
- $\text{height} = 40 \times \tan(35^\circ)$
- $\tan(35^\circ) \approx 0.7002$
- $\text{height} = 40 \times 0.7002 \approx 28.01$ ft

Problem 10:
A 15 ft ladder leans against a wall. The top of the ladder reaches 9 ft up the wall. What is the angle the ladder makes with the ground?
- We know the opposite side (9 ft) and the hypotenuse (15 ft). We need to find the angle $\theta$.
- Use sine: $\sin(\theta) = \frac{9}{15} = 0.6$
- $\theta = \arcsin(0.6)$
- $\theta \approx 36.87^\circ$

Final Answer:
1. Approximately 18.33 ft
2. Approximately 53.13°
3. Approximately 86.6 ft
4. Approximately 84.85 ft
5. Approximately 11.55 ft
6. Approximately 3.42 ft
7. Approximately 35.75 ft
8. Approximately 65.38°
9. Approximately 28.01 ft
10. Approximately 36.87°