How to Solve: Sine, Cosine, Tangent Problems

How to Solve: Sine, Cosine, Tangent Problems

A Complete Guide for Students & Teachers

Introduction

"Master sine, cosine, and tangent, and you’ll solve real-world problems—from measuring skyscrapers to acing your trigonometry exam in under 60 seconds per question."

What You Need To Know First

Before diving into SOH-CAH-TOA, ensure you understand: 1. Right-angled triangles – A triangle with one 90° angle. 2. Opposite, adjacent, and hypotenuse sides – Relative to the angle you’re working with. 3. Basic algebra – Rearranging equations to solve for unknowns.

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Identify the Given Information

• Look for:
• An angle (θ) and one side length.
• Two side lengths (to find an angle).
• Label the triangle with:
• Opposite (O) – Across from the angle.
• Adjacent (A) – Next to the angle (not hypotenuse).
• Hypotenuse (H) – Longest side (always opposite 90°).

Step 2: Choose the Correct Ratio (SOH-CAH-TOA)

• If you have O and H → Use SINE (sin).
• If you have A and H → Use COSINE (cos).
• If you have O and A → Use TANGENT (tan).

Step 3: Write the Equation

• Plug the known values into the formula.
• Example: If θ = 30°, Opposite = 5, find Hypotenuse.
• sin(30°) = 5 / H → H = 5 / sin(30°)

Step 4: Solve for the Unknown

• Rearrange the equation.
• Use a calculator (ensure it’s in degree mode).
• Example: H = 5 / 0.5 = 10

Step 5: Check Your Answer

• Does the side length make sense?
• Hypotenuse must be the longest side.
• Angles must be between 0° and 90°.

Worked Examples

Example 1 – Basic (Find a Side)

Problem: In a right-angled triangle, angle A = 35°, the hypotenuse is 12 cm. Find the length of the side opposite angle A.

Solution: 1. Identify: Angle = 35°, Hypotenuse (H) = 12 cm, Opposite (O) = ? 2. Choose ratio: We have O and H → Use SINE (sin). 3. Write equation: sin(35°) = O / 12 4. Solve for O: O = 12 × sin(35°) 5. Calculate: O = 12 × 0.5736 ≈ 6.88 cm

What we did and why: - We used sin because we had the opposite side and hypotenuse. - Multiplied the hypotenuse by sin(θ) to find the opposite side.

Example 2 – Medium (Find an Angle)

Problem: In a right-angled triangle, the side opposite angle B is 7 cm, and the adjacent side is 10 cm. Find angle B.

Solution: 1. Identify: Opposite (O) = 7 cm, Adjacent (A) = 10 cm, Angle B = ? 2. Choose ratio: We have O and A → Use TANGENT (tan). 3. Write equation: tan(B) = 7 / 10 4. Solve for B: B = tan⁻¹(7 / 10) 5. Calculate: B ≈ 34.99°

What we did and why: - We used tan because we had the opposite and adjacent sides. - Used tan⁻¹ (inverse tangent) to find the angle.

Example 3 – Exam Style (Word Problem)

Problem: A ladder leans against a wall at a 60° angle. If the base of the ladder is 4 m from the wall, how high up the wall does the ladder reach?

Solution: 1. Draw the triangle:
- Wall = Opposite (O)
- Ground = Adjacent (A) = 4 m
- Ladder = Hypotenuse (H)
- Angle = 60° 2. Choose ratio: We have A and need O → Use TANGENT (tan). 3. Write equation: tan(60°) = O / 4 4. Solve for O: O = 4 × tan(60°) 5. Calculate: O = 4 × 1.732 ≈ 6.93 m

What we did and why: - We used tan because we had the adjacent side and needed the opposite. - Multiplied the adjacent side by tan(θ) to find the height.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—tonight, before your exam, here’s what you need to remember:

1. Label your triangle: Opposite, adjacent, hypotenuse—always relative to the angle you’re using.
2. Pick the right ratio: SOH-CAH-TOA. If you have two sides, use the one that matches.
3. Write the equation: Plug in the numbers, then solve.
4. Check your calculator: DEGREE MODE! No radians.
5. Double-check: Does the answer make sense? Hypotenuse is always the longest side.

You’ve got this. One question at a time, stay calm, and trust the steps. Good luck!