Trigonometry worksheet with problems on angles of elevation and depression, showing step-by-step solutions using tangent functions.

Let’s solve each problem step by step. We’ll use trigonometry — specifically, the tangent function — because we’re dealing with right triangles and angles of elevation or depression.

Remember:
In a right triangle,
tan(angle) = opposite / adjacent

We’ll label:
- “Opposite” as the side across from the angle (usually the height we’re trying to find).
- “Adjacent” as the side next to the angle (the horizontal distance given).

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Problem 1:

> The angle of elevation from a point 20 feet from the base of a tree on level ground to the top of the tree is 30 degrees. Find the height of the tree to the nearest tenth.

Step 1: Draw a right triangle.
- Angle at ground = 30°
- Adjacent side (ground) = 20 ft
- Opposite side (tree height) = x (what we want)

Step 2: Use tan(30°) = opposite / adjacent → tan(30°) = x / 20

Step 3: Solve for x:
x = 20 × tan(30°)

Step 4: Calculate tan(30°).
tan(30°) ≈ 0.57735

So,
x ≈ 20 × 0.57735 ≈ 11.547

Step 5: Round to nearest tenth → 11.5 ft

✔ Final Answer for #1: 11.5 ft

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Problem 2:

> A person measures the angle of depression from the top of a wall to a point on the ground. The point is located on level ground 62 feet from the base of the wall and the angle of depression is 52°. How high is the wall, to the nearest tenth of a foot?

Important note: Angle of depression from the top equals the angle of elevation from the bottom (because they are alternate interior angles). So we can treat this as an angle of elevation of 52° from the ground point to the top of the wall.

Step 1: Right triangle.
- Angle at ground = 52°
- Adjacent side (ground) = 62 ft
- Opposite side (wall height) = x

Step 2: tan(52°) = x / 62

Step 3: Solve for x:
x = 62 × tan(52°)

Step 4: tan(52°) ≈ 1.2799

x ≈ 62 × 1.2799 ≈ 79.3538

Step 5: Round to nearest tenth → 79.4 ft

✔ Final Answer for #2: 79.4 ft

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Problem 3:

> John wants to measure the height of a tree. He walks exactly 100 feet from the base of the tree and looks up. The angle from the ground to the top of the tree is 33°. How tall is the tree to the nearest foot?

Step 1: Right triangle.
- Angle at ground = 33°
- Adjacent side = 100 ft
- Opposite side (tree height) = x

Step 2: tan(33°) = x / 100

Step 3: Solve for x:
x = 100 × tan(33°)

Step 4: tan(33°) ≈ 0.6494

x ≈ 100 × 0.6494 ≈ 64.94

Step 5: Round to nearest foot → 65 ft

✔ Final Answer for #3: 65 ft

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Final Answer:
1) 11.5 ft
2) 79.4 ft
3) 65 ft