Geometry
Using Trigonometry to Find Height of Objects or Length of Shadow
Trigonometry solves real-world problems involving angles of elevation or depression. Set up a right triangle, identify the known angle and side, then use sin, cos, or tan to find the unknown.
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Worked Examples
Follow along with these step-by-step examples. Take your time — there's no rush!
Problem
You stand 50 feet from a tree. The angle of elevation to the top is 35°. How tall is the tree?
Step-by-Step Solution
You know the adjacent side (50 ft) and want the opposite side (height).
Use tan: tan 35° = height/50
height = 50 × tan 35° ≈ 50 × 0.7002 ≈ 35 feet
Answer
≈ 35 feet
Problem
A 20-foot ladder leans against a wall at a 60° angle with the ground. How high does it reach?
Step-by-Step Solution
Hypotenuse = 20, angle = 60°, want opposite (height).
sin 60° = height/20
height = 20 × sin 60° = 20 × 0.866 ≈ 17.3 feet
Answer
≈ 17.3 feet
Problem
A building is 100 feet tall. The angle of elevation from a point on the ground is 45°. How far away is the point?
Step-by-Step Solution
Opposite = 100, angle = 45°, want adjacent (distance).
tan 45° = 100/distance → 1 = 100/distance
distance = 100 feet
Answer
100 feet away
Problem
A ramp rises 4 feet over a horizontal distance of 20 feet. What is the angle of inclination?
Step-by-Step Solution
tan θ = opposite/adjacent = 4/20 = 0.2
θ = tan⁻¹(0.2) ≈ 11.3°
Answer
≈ 11.3°
Problem
From the top of a 60-foot cliff, the angle of depression to a boat is 30°. How far is the boat from the base of the cliff?
Step-by-Step Solution
Angle of depression = angle of elevation from boat = 30°.
tan 30° = 60/distance → distance = 60/tan 30°
distance = 60/0.577 ≈ 103.9 feet
Answer
≈ 103.9 feet
Your Turn — Practice Problems
Try all 5 problems on your own first. Write out your work — that's how it sticks!
💡 Tip: Don't peek at the answers until you've genuinely tried each one.
Angle of elevation 30°, distance from base = 40 ft. Find height.
Ladder 15 ft, angle 50°. How high does it reach?
Building 80 ft tall, angle of elevation 40°. How far away?
Angle of elevation 60°, height = 30 ft. Find distance from base.
A 10-ft pole casts a shadow. Sun angle = 45°. How long is the shadow?
Angle of elevation 45°, distance from base = 50 ft. Find height.
Ladder 20 ft, angle 30°. How high does it reach?
A tree is 40 ft tall. Angle of elevation from ground = 60°. How far away?
Angle of depression 45° from a 100-ft cliff. How far is the boat?
Ladder 10 ft, angle 60°. How far is the base from the wall?
A ramp rises 3 ft over 12 ft horizontal. What is the angle?
Angle of elevation 30°, height = 15 ft. Find distance from base.
Finished all 12? Give yourself a pat on the back — then check your work!
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