How to Solve: Basic Trigonometric Ratios

How to Solve: Basic Trigonometric Ratios

Introduction

"Mastering trig ratios lets you calculate the height of a tree, the angle of a ramp, or even the distance to a star—all without leaving your desk. And on exam day, it’s the difference between a quick 5-mark question and a frustrating 10-minute struggle."

What You Need To Know First

Before diving into trig ratios, ensure you understand: 1. Right-angled triangles: A triangle with one 90° angle. 2. Labeling sides: Opposite, adjacent, and hypotenuse relative to a given angle. 3. Pythagoras’ theorem: a² + b² = c² (used to find missing sides).

If any of these are unclear, review them first—trig ratios build directly on them.

Key Vocabulary

Formulas To Know

1. SOH-CAH-TOA (The Core Trig Ratios)

MEMORISE THIS – Examiners expect you to recall this instantly.

Mnemonic: "Some Old Horses Can Always Hear Their Owners Approach" (SOH-CAH-TOA).

2. Pythagoras’ Theorem (For Finding Missing Sides)

Given on exam sheet (but you should know it anyway). Formula: a² + b² = c² - a and b = legs (shorter sides). - c = hypotenuse (longest side).

Step-by-Step Method

How to Solve Any Basic Trig Ratio Problem

1. Identify the given angle and sides.
2. Circle the angle (θ) you’re working with.

3. Label the sides relative to θ: opposite, adjacent, hypotenuse.

4. Choose the correct ratio (SOH-CAH-TOA).

5. If you have opposite + hypotenuse → use sine.
6. If you have adjacent + hypotenuse → use cosine.

7. If you have opposite + adjacent → use tangent.

8. Write the formula with the known values.

9. Example: If θ = 30°, opposite = 5, and you need the hypotenuse:
   sin(30°) = 5 / hypotenuse.

10. Rearrange to solve for the unknown.

11. Multiply or divide to isolate the missing side/angle.

12. Example: hypotenuse = 5 / sin(30°) = 5 / 0.5 = 10.

13. Check your answer makes sense.

14. The hypotenuse must be the longest side.
15. Angles in a triangle add up to 180°.

Worked Example Using the Steps

Problem: In a right-angled triangle, θ = 40°, the adjacent side is 6 cm. Find the opposite side.

1. Identify the angle and sides:
2. Angle: θ = 40°.
3. Adjacent = 6 cm.

4. Opposite = ? (what we’re finding).

5. Choose the ratio:

6. We have adjacent + opposite → use tangent.

7. tan(θ) = opposite / adjacent.

8. Write the formula:

9. tan(40°) = opposite / 6.

10. Rearrange to solve:

11. opposite = 6 × tan(40°).
12. tan(40°) ≈ 0.8391 (use calculator).

13. opposite ≈ 6 × 0.8391 ≈ 5.03 cm.

14. Check:

15. The opposite side (5.03 cm) is shorter than the hypotenuse (which would be longer than 6 cm) → makes sense.

Worked Examples

Example 1 – Basic (Finding a Side)

Problem: In a right-angled triangle, θ = 25°, hypotenuse = 10 m. Find the opposite side.

1. Label the sides:
2. Angle: 25°.
3. Hypotenuse = 10 m.

4. Opposite = ?.

5. Choose ratio:

6. Opposite + hypotenuse → sine.

7. sin(25°) = opposite / 10.

8. Solve:

9. opposite = 10 × sin(25°).
10. sin(25°) ≈ 0.4226.
11. opposite ≈ 10 × 0.4226 ≈ 4.23 m.

What we did and why: We used sine because we had the hypotenuse and needed the opposite side. The calculator gave us the decimal for sin(25°), and multiplying by 10 gave the missing side.

Example 2 – Medium (Finding an Angle)

Problem: In a right-angled triangle, the opposite side is 7 cm, and the hypotenuse is 12 cm. Find θ.

1. Label the sides:
2. Opposite = 7 cm.
3. Hypotenuse = 12 cm.

4. Angle = ?.

5. Choose ratio:

6. Opposite + hypotenuse → sine.

7. sin(θ) = 7 / 12.

8. Solve for θ:

9. θ = sin⁻¹(7 / 12) (use inverse sine on calculator).
10. 7 / 12 ≈ 0.5833.
11. θ ≈ sin⁻¹(0.5833) ≈ 35.45°.

What we did and why: We used inverse sine because we had the ratio but needed the angle. The calculator’s sin⁻¹ function gives the angle when you know the ratio.

Example 3 – Exam Style (Disguised Problem)

Problem: A ladder leans against a wall at a 65° angle. The base of the ladder is 2 m from the wall. How high up the wall does the ladder reach? (Give your answer to 2 decimal places.)

1. Draw the triangle:
2. Wall = opposite side.
3. Ground = adjacent side (2 m).
4. Ladder = hypotenuse.

5. Angle = 65°.

6. Choose ratio:

7. Adjacent + opposite → tangent.

8. tan(65°) = opposite / 2.

9. Solve:

10. opposite = 2 × tan(65°).
11. tan(65°) ≈ 2.1445.
12. opposite ≈ 2 × 2.1445 ≈ 4.29 m.

What we did and why: This is a real-world problem, but it’s just a trig ratio in disguise. We used tangent because we had the adjacent side and needed the opposite side. The answer is the height the ladder reaches.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Trig ratios are just SOH-CAH-TOA: - Sine = opposite over hypotenuse. - Cosine = adjacent over hypotenuse. - Tangent = opposite over adjacent.

Step 1: Label your triangle—opposite, adjacent, hypotenuse—relative to the angle you’re using. Step 2: Pick the right ratio based on the sides you have. Step 3: Plug in the numbers, solve, and check your answer makes sense—hypotenuse is always the longest side!

Watch out for: - Mixing up opposite and adjacent. - Forgetting inverse trig when finding angles. - Calculator mode (always DEG for exams).

Practice 3 problems tonight—one for each ratio—and you’ll own this on exam day. You’ve got this!