How to Solve: Angle of Elevation

How to Solve: Angle of Elevation

Introduction

"If you’ve ever wondered how tall a tree is without climbing it—or how far a drone is from the ground—mastering the angle of elevation is your secret weapon. It’s a guaranteed 3-5 marks on every trigonometry exam, and it’s easier than you think."

What You Need To Know First

Before diving into angle of elevation, make sure you understand: 1. Right-angled triangles: Know the sides (opposite, adjacent, hypotenuse) and how they relate to angles. 2. Basic trigonometric ratios: SOH-CAH-TOA (sine, cosine, tangent) and how to use them. 3. How to solve for unknown sides or angles using inverse trig functions (e.g., tan⁻¹).

If any of these feel shaky, pause here and review them first.

Key Vocabulary

Formulas To Know

1. Tangent Ratio (Most Common for Angle of Elevation)

Formula: [ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} ] Variables: - (\theta) = angle of elevation (in degrees) - opposite = height of the object (vertical side) - adjacent = distance from the observer to the base of the object (horizontal side)

Memorise this? ✅ MEMORISE THIS (SOH-CAH-TOA is essential).

2. Solving for the Angle (Inverse Tangent)

Formula: [ \theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) ] Variables: - Same as above, but now you’re solving for (\theta).

Memorise this? ✅ MEMORISE THIS (you’ll use this often).

3. Solving for a Side (Rearranged Tangent)

Formula: [ \text{opposite} = \text{adjacent} \times \tan(\theta) ] or [ \text{adjacent} = \frac{\text{opposite}}{\tan(\theta)} ]

Memorise this? ❌ Given on exam sheet (but you should know how to rearrange it).

Step-by-Step Method

Follow these steps for EVERY angle of elevation problem.

1. Draw a diagram.
2. Sketch a right-angled triangle.
3. Label the horizontal line (ground), vertical line (height of object), and line of sight.

4. Mark the angle of elevation ((\theta)) at the observer’s position.

5. Identify the known and unknown values.

6. What’s given? (e.g., height, distance, angle)

7. What are you solving for? (e.g., angle, height, distance)

8. Label the sides of the triangle.

9. Opposite = height of the object (vertical side).
10. Adjacent = distance from observer to object (horizontal side).

11. Hypotenuse = line of sight (rarely needed for angle of elevation).

12. Choose the correct trig ratio.

13. If you have opposite and adjacent, use tangent.
14. If you have hypotenuse and opposite, use sine.

15. If you have hypotenuse and adjacent, use cosine.

16. Plug values into the formula and solve.

17. For angles: Use (\tan^{-1}).

18. For sides: Rearrange the formula.

19. Check units and reasonableness.

20. Angles should be between 0° and 90°.

21. Heights/distances should make sense (e.g., a tree isn’t 500m tall).

22. Write a clear final answer with units.

23. Example: "The angle of elevation is 35°."

Worked Example Using the Steps

Problem: A person stands 20m away from a tree. The angle of elevation to the top of the tree is 40°. How tall is the tree?

Solution: 1. Draw a diagram.
- Right-angled triangle: horizontal = 20m, vertical = height (h), angle = 40°.

1. Identify known and unknown.
2. Known: adjacent = 20m, angle = 40°.

3. Unknown: opposite (height, h).

4. Label sides.

5. Opposite = h (height).

6. Adjacent = 20m.

7. Choose trig ratio.

8. We have opposite and adjacent → tangent.

9. (\tan(40°) = \frac{h}{20})

10. Solve for h.

11. (h = 20 \times \tan(40°))
12. (h = 20 \times 0.8391)

13. (h ≈ 16.78m)

14. Check reasonableness.

15. 16.78m is a reasonable height for a tree.

16. Final answer:

17. "The tree is 16.8m tall (to 1 decimal place)."

Worked Examples

Example 1 - Basic

Problem: A ladder leans against a wall. The foot of the ladder is 3m from the wall, and the angle of elevation is 60°. How high up the wall does the ladder reach?

Solution: 1. Diagram: Right-angled triangle, horizontal = 3m, angle = 60°, vertical = h. 2. Known: adjacent = 3m, angle = 60°. 3. Unknown: opposite (height, h). 4. Use tangent: (\tan(60°) = \frac{h}{3}) 5. Solve: (h = 3 \times \tan(60°) = 3 \times 1.732 ≈ 5.20m) 6. Check: 5.20m is reasonable for a ladder. 7. Answer: "The ladder reaches 5.2m up the wall."

What we did and why: - We used tangent because we had the adjacent side and needed the opposite. - The angle was given, so we didn’t need inverse trig.

Example 2 - Medium

Problem: A drone is flying at a height of 80m. The angle of elevation from a person on the ground to the drone is 25°. How far is the person from the point directly below the drone?

Solution: 1. Diagram: Right-angled triangle, vertical = 80m, angle = 25°, horizontal = d. 2. Known: opposite = 80m, angle = 25°. 3. Unknown: adjacent (distance, d). 4. Use tangent: (\tan(25°) = \frac{80}{d}) 5. Rearrange: (d = \frac{80}{\tan(25°)} ≈ \frac{80}{0.4663} ≈ 171.6m) 6. Check: 171.6m is a reasonable distance. 7. Answer: "The person is 172m away (to 3 significant figures)."

What we did and why: - We rearranged the tangent formula because we needed the adjacent side. - The angle was given, so we used (\tan(25°)) directly.

Example 3 - Exam Style

Problem: A surveyor stands 50m from the base of a building. The angle of elevation to the top of the building is 32°. The surveyor’s eye level is 1.6m above the ground. What is the total height of the building?

Solution: 1. Diagram: Two parts:
- Right-angled triangle: horizontal = 50m, angle = 32°, vertical = h (height above eye level).
- Eye level = 1.6m. 2. Known: adjacent = 50m, angle = 32°, eye level = 1.6m. 3. Unknown: total height = h + 1.6m. 4. Use tangent: (\tan(32°) = \frac{h}{50}) 5. Solve: (h = 50 \times \tan(32°) ≈ 50 \times 0.6249 ≈ 31.24m) 6. Total height = 31.24m + 1.6m = 32.84m 7. Check: 32.84m is reasonable for a building. 8. Answer: "The building is 32.8m tall (to 1 decimal place)."

What we did and why: - We had to add the eye level to the height from the triangle. - The problem tested two-step thinking, which is common in exams.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. Angle of elevation is just a fancy way of saying ‘the angle you look up at something.’ Here’s how to crush it on exam day:

1. Draw a right-angled triangle every single time. Label the horizontal line, vertical line, and the angle.
2. Pick the right ratio: If you have opposite and adjacent, use tangent. If you need the angle, use inverse tangent.
3. Watch for traps: Eye level? Add it. Multiple steps? Solve one part at a time.
4. Check your answer: Does it make sense? A 5° angle shouldn’t give you a 500m tall building.
5. Practice with real exam questions—this is how you’ll spot the tricks.

You’ve got this. Now go ace that exam!