How to Solve: Trigonometric Equations & Proving Identities

How to Solve: Trigonometric Equations & Proving Identities

GCSE / A-Level Maths

Introduction

"Mastering trig equations and identities doesn’t just get you 10–15 marks on your exam—it unlocks real-world problems like predicting tides, designing roller coasters, or even calculating how far a rocket travels. Miss this, and you’re leaving easy marks on the table. Let’s fix that."

What You Need To Know First

Before diving in, you must already understand: 1. Basic trig ratios (sin, cos, tan) – How to find them in right-angled triangles and on the unit circle. 2. Trig graphs – The shapes of y = sin x, y = cos x, and y = tan x, including their key points (e.g., max/min, asymptotes). 3. Algebraic manipulation – Expanding brackets, factorising, and solving linear/quadratic equations.

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

Key Vocabulary

Formulas To Know

1. Pythagorean Identities (MEMORISE THIS)

• sin²θ + cos²θ ≡ 1
• What it means: For any angle θ, squaring sin and cos and adding them gives 1.
• 1 + tan²θ ≡ sec²θ
• What it means: sec θ = 1/cos θ (reciprocal identity).
• 1 + cot²θ ≡ cosec²θ
• What it means: cosec θ = 1/sin θ, cot θ = 1/tan θ.

2. Double-Angle Formulas (MEMORISE THIS)

• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos²θ – sin²θ (or 2 cos²θ – 1 or 1 – 2 sin²θ)
• tan(2θ) = (2 tan θ) / (1 – tan²θ)

3. Compound Angle Formulas (Given on exam sheet)

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

4. General Solutions (MEMORISE THIS)

For sin x = k: - x = arcsin(k) + 360°n or x = 180° – arcsin(k) + 360°n (where n is any integer).

For cos x = k: - x = arccos(k) + 360°n or x = –arccos(k) + 360°n.

For tan x = k: - x = arctan(k) + 180°n.

Step-by-Step Method

Part 1: Solving Trigonometric Equations

Step 1: Isolate the trig function - Get sin x, cos x, or tan x by itself. - Example: 3 sin x – 1 = 0 → sin x = 1/3.

Step 2: Find the principal value - Use arcsin, arccos, or arctan to find the first angle in 0°–360°. - Example: sin x = 1/3 → x = arcsin(1/3) ≈ 19.5°.

Step 3: Use the CAST diagram to find all solutions in 0°–360° - CAST rules: - Sine is positive in Second quadrant (180° – θ). - All (sin, cos, tan) are positive in All first quadrant (θ). - Tangent is positive in Third quadrant (180° + θ). - Cosine is positive in Chird quadrant (360° – θ). - Example: sin x = 1/3 → x = 19.5° (1st quadrant) or x = 180° – 19.5° = 160.5° (2nd quadrant).

Step 4: Write the general solution - Add + 360°n to each solution. - Example: x = 19.5° + 360°n or x = 160.5° + 360°n.

Step 5: Check the question’s range - If the question asks for 0° ≤ x ≤ 360°, list all solutions in that range. - If it asks for –180° ≤ x ≤ 180°, adjust accordingly.

Part 2: Proving Trigonometric Identities

Step 1: Start with the more complex side - Pick the side with more terms or operations (e.g., sin²x + cos²x vs. 1).

Step 2: Use known identities to rewrite terms - Replace sin²x with 1 – cos²x (or vice versa). - Use double-angle formulas if needed.

Step 3: Simplify step-by-step - Expand brackets, combine fractions, or factorise. - Example: sin x / (1 – cos x) = (1 + cos x) / sin x → Multiply numerator and denominator by (1 + cos x).

Step 4: Show both sides are equal - Keep simplifying until both sides match. - Never assume the identity is true—always prove it!

Worked Examples

Example 1 – Basic: Solve 2 cos x = √3 for 0° ≤ x ≤ 360°

Step 1: Isolate cos x → cos x = √3 / 2. Step 2: Principal value → x = arccos(√3 / 2) = 30°. Step 3: CAST diagram → Cosine is positive in 1st and 4th quadrants. - 1st quadrant: x = 30°. - 4th quadrant: x = 360° – 30° = 330°. Step 4: General solution → x = 30° + 360°n or x = 330° + 360°n. Step 5: Range is 0°–360° → x = 30° or 330°.

What we did and why: We isolated cos x, found the principal value, used the CAST diagram to get all solutions in the range, and checked the question’s limits.

Example 2 – Medium: Solve sin(2x) = √3 / 2 for 0° ≤ x ≤ 360°

Step 1: Let θ = 2x → sin θ = √3 / 2. Step 2: Principal value → θ = arcsin(√3 / 2) = 60°. Step 3: CAST diagram → Sine is positive in 1st and 2nd quadrants. - 1st quadrant: θ = 60°. - 2nd quadrant: θ = 180° – 60° = 120°. Step 4: General solution → θ = 60° + 360°n or θ = 120° + 360°n. Step 5: Substitute back θ = 2x → 2x = 60° + 360°n or 2x = 120° + 360°n. - x = 30° + 180°n or x = 60° + 180°n. Step 6: Find x in 0°–360°: - For n = 0: x = 30° or 60°. - For n = 1: x = 210° or 240°. - For n = 2: x = 390° (out of range).

Solutions: x = 30°, 60°, 210°, 240°.

What we did and why: We used substitution to simplify the equation, solved for θ, then converted back to x. The 180°n comes from halving the 360°n period.

Example 3 – Exam-Style: Prove (1 – cos x)(1 + cos x) = sin²x

Step 1: Start with the left side (more complex). Step 2: Expand brackets → 1 – cos²x. Step 3: Use Pythagorean identity → sin²x + cos²x = 1 → 1 – cos²x = sin²x. Step 4: Right side is sin²x → Both sides match.

What we did and why: We expanded the left side, used a known identity to rewrite it, and showed it equals the right side. This proves the identity is true for all x.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute cheat sheet for trig equations and identities. For equations: 1. Isolate the trig function (sin/cos/tan). 2. Find the principal value with arcsin/arccos/arctan. 3. Use the CAST diagram to get all solutions in 0°–360°. 4. Add 360°n for the general solution, then check the range. 5. For double angles, substitute θ = 2x first, solve, then convert back.

For identities: 1. Start with the messier side. 2. Use identities (Pythagorean, double-angle) to rewrite terms. 3. Simplify step-by-step until both sides match. 4. Never assume—always prove!

Common traps? - Forgetting the second solution (CAST diagram!). - Ignoring the range (examiners love this). - Dividing by trig functions (factorise instead!).

You’ve got this. Now go smash those marks!