How to Solve: Trigonometric Word Problems

How to Solve: Trigonometric Word Problems

Introduction

"Mastering trigonometric word problems means you can calculate the height of a tree without climbing it, design a ramp for a wheelchair, or even predict the path of a rocket—all in under 5 minutes on your exam."

What You Need To Know First

Before tackling trigonometric word problems, you must already understand: 1. Basic trigonometric ratios (SOH-CAH-TOA) – How to find sine, cosine, and tangent in right-angled triangles. 2. Pythagoras’ theorem – Used to find missing sides when two sides are known. 3. Angle of elevation/depression – The angle between the horizontal line and the line of sight (up or down).

If any of these are unclear, review them first—this guide assumes you’re solid on them.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Follow these 6 steps for every trigonometric word problem:

1. Read the problem carefully.
2. Underline key numbers (distances, angles, heights).

3. Circle words like "elevation," "depression," "bearing," or "shadow."

4. Draw a diagram.

5. Sketch the scenario as a right-angled triangle (even if it’s not obvious).
6. Label all known sides and angles.

7. Mark the unknown side/angle with a question mark.

8. Identify the trigonometric ratio to use.

9. Ask: "Which sides do I know, and which do I need?"

10. If you have:

    • Opposite & Hypotenuse → sin θ
    • Adjacent & Hypotenuse → cos θ
    • Opposite & Adjacent → tan θ

11. Write the equation.

12. Plug the known values into the correct ratio (SOH-CAH-TOA).

13. Example: If you need the opposite side and know the angle and adjacent, use tan θ = opposite / adjacent.

14. Solve for the unknown.

15. Rearrange the equation to isolate the unknown.

16. Use inverse trig functions (sin⁻¹, cos⁻¹, tan⁻¹) if finding an angle.

17. Check your answer.

18. Does it make sense? (e.g., a tree can’t be 5 cm tall.)
19. Did you use the correct units? (meters, degrees, etc.)

Worked Example Using the Steps

Problem: A ladder leans against a wall. The foot of the ladder is 2 meters from the wall, and the ladder makes a 60° angle with the ground. How long is the ladder?

Step 1: Read the problem. - Known: Distance from wall = 2 m, angle = 60°. - Unknown: Length of ladder (hypotenuse).

Step 2: Draw a diagram.

       /|

      / |

     /  |
  L /   | H

   /    |
  /60°  |
 /______|

   2 m

• L = ladder (hypotenuse)
• 2 m = adjacent side to 60°
• H = height (opposite side)

Step 3: Identify the ratio. - We know the adjacent side (2 m) and need the hypotenuse (L). - Use cos θ = adjacent / hypotenuse.

Step 4: Write the equation. - cos 60° = 2 / L

Step 5: Solve for L. - Rearrange: L = 2 / cos 60° - cos 60° = 0.5 - L = 2 / 0.5 = 4 meters

Step 6: Check the answer. - A 4 m ladder leaning at 60° with a 2 m base is reasonable.

Worked Examples

Example 1 - Basic: Finding a Missing Side

Problem: A kite string is 50 meters long and makes a 35° angle with the ground. How high is the kite?

Solution: 1. Diagram:
/|
/ |
50/ | H
/ |
/35° | /_____|
- Hypotenuse = 50 m
- Angle = 35°
- Opposite side = height (H)

1. Ratio: sin θ = opposite / hypotenuse → sin 35° = H / 50

2. Equation: H = 50 × sin 35°

3. Calculate: sin 35° ≈ 0.5736 → H ≈ 50 × 0.5736 ≈ 28.68 meters

What we did and why: - We used sin θ because we had the hypotenuse and needed the opposite side (height). - Always label the diagram first—it makes choosing the ratio easier.

Example 2 - Medium: Angle of Depression

Problem: From the top of a 20-meter cliff, the angle of depression to a boat is 25°. How far is the boat from the base of the cliff?

Solution: 1. Diagram:
Cliff top
---------
| 25°\
| \
| \
20 m \
| \
| \
---------
Boat
- The angle of depression (25°) is outside the triangle.
- The angle inside the triangle is also 25° (alternate angles).

1. Ratio: tan θ = opposite / adjacent → tan 25° = 20 / distance

2. Equation: distance = 20 / tan 25°

3. Calculate: tan 25° ≈ 0.4663 → distance ≈ 20 / 0.4663 ≈ 42.89 meters

What we did and why: - The angle of depression is equal to the angle inside the triangle (alternate angles). - We used tan θ because we had the opposite (cliff height) and needed the adjacent (distance to boat).

Example 3 - Exam Style: Bearing Problem

Problem: A ship sails 10 km on a bearing of 030° from port. How far east of the port is the ship?

Solution: 1. Diagram:
N
|
| 30°
--------> E
/ \
/ \ 10/ \ x
/ \ ---------
Port
- Bearing 030° means 30° east of north.
- The eastward distance (x) is the adjacent side to the 30° angle.

1. Ratio: cos θ = adjacent / hypotenuse → cos 30° = x / 10

2. Equation: x = 10 × cos 30°

3. Calculate: cos 30° ≈ 0.8660 → x ≈ 10 × 0.8660 ≈ 8.66 km

What we did and why: - Bearings are measured from north, so we drew the angle from the vertical. - We used cos θ because we needed the adjacent side (eastward distance) and had the hypotenuse (10 km).

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in for your exam tomorrow. Trigonometric word problems are just SOH-CAH-TOA in disguise. Here’s the game plan:

1. Read, underline, draw – Sketch a right-angled triangle and label everything.
2. Pick your ratio – Opposite/hypotenuse? Sin. Adjacent/hypotenuse? Cos. Opposite/adjacent? Tan.
3. Write the equation – Plug in the numbers and solve.
4. Check your answer – Does it make sense? Did you use the right units?

Watch out for: - Angles of depression – They’re equal to the angle inside the triangle. - Bearings – Start from north and go clockwise. - Extra info – Don’t let it distract you.

You’ve got this. Now go practice two problems tonight—one basic, one with a twist. See you in the exam!