How to Solve: Trigonometry (Radians, Sec, Cosec, Cot, Compound/Double Angle Identities, R-Form, Small Angles)

How to Solve: Trigonometry (Radians, Sec, Cosec, Cot, Compound/Double Angle Identities, R-Form, Small Angles)

GCSE / A-Level Maths

Introduction

"Mastering these trig identities doesn’t just get you marks—it unlocks the hardest A-Level questions, like proving identities worth 6+ marks or solving equations where others get stuck. On the Edexcel A-Level, this topic alone can add 10-15% to your final grade. Let’s break it down so you never lose a mark again."

What You Need To Know First

1. Basic trig ratios (sin, cos, tan) – You must know these for any angle, including special triangles (30°, 45°, 60°).
2. Pythagorean identities – e.g., sin²θ + cos²θ = 1.
3. Degrees vs. radians – How to convert between them (π radians = 180°).

Key Vocabulary

Formulas To Know

1. Reciprocal Identities

• secθ = 1/cosθ (MEMORISE THIS)
• cosecθ = 1/sinθ (MEMORISE THIS)
• cotθ = 1/tanθ = cosθ/sinθ (MEMORISE THIS)

2. Pythagorean Identities (Extended)

• 1 + tan²θ = sec²θ (MEMORISE THIS – derived from sin²θ + cos²θ = 1)
• 1 + cot²θ = cosec²θ (MEMORISE THIS)

3. Compound Angle Formulas

• sin(A ± B) = sinA cosB ± cosA sinB (MEMORISE THIS)
• cos(A ± B) = cosA cosB ∓ sinA sinB (MEMORISE THIS)
• tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB) (MEMORISE THIS)

4. Double Angle Formulas

• sin(2θ) = 2sinθ cosθ (MEMORISE THIS)
• cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ (MEMORISE THIS – all 3 forms!)
• tan(2θ) = 2tanθ / (1 – tan²θ) (MEMORISE THIS)

5. R-Form (a sinθ + b cosθ)

• a sinθ + b cosθ = R sin(θ + α), where:
• R = √(a² + b²) (MEMORISE THIS)
• tanα = b/a (MEMORISE THIS – α is the phase shift)

6. Small Angle Approximations (in radians!)

• sinθ ≈ θ (MEMORISE THIS – only for θ close to 0)
• tanθ ≈ θ (MEMORISE THIS)
• cosθ ≈ 1 – θ²/2 (MEMORISE THIS)

Step-by-Step Method

How to Solve Any Trig Problem (Step-by-Step)

1. Identify the type of problem – Is it an identity to prove? An equation to solve? A simplification?
2. Convert to radians if needed – Check if the question specifies radians (e.g., "θ is small").
3. Rewrite in terms of sin/cos – If sec, cosec, or cot appear, replace them (e.g., secθ = 1/cosθ).
4. Use identities – Pick the right formula (compound, double angle, or Pythagorean).
5. Simplify – Factorise, cancel terms, or combine fractions.
6. Check for exact values – If angles like 30°, 45°, 60° appear, use exact values.
7. Solve or prove – For equations, find all solutions in the given range. For proofs, show both sides are equal.
8. Verify – Plug in a value (e.g., θ = 30°) to check your answer.

Worked Example (Using the Steps)

Problem: Prove that cosec²θ – cot²θ = 1.

Step 1: Identify – This is a proof, so we’ll rewrite everything in terms of sin/cos. Step 2: No radians needed here. Step 3: Rewrite cosec and cot: - cosec²θ = 1/sin²θ - cot²θ = cos²θ/sin²θ Step 4: Substitute: - 1/sin²θ – cos²θ/sin²θ = (1 – cos²θ)/sin²θ Step 5: Simplify numerator using Pythagorean identity: - 1 – cos²θ = sin²θ - So, (sin²θ)/sin²θ = 1 Step 6: No exact values needed. Step 7: Both sides equal 1, so the identity is proven. Step 8: Verify with θ = 30°: - cosec²(30°) = 4, cot²(30°) = 3 → 4 – 3 = 1 ✔️

What we did and why: We rewrote everything in terms of sin/cos to use the Pythagorean identity. This is the standard approach for proving trig identities—simplify one side to match the other.

Worked Examples

Example 1 – Basic: Solve secθ = 2 for 0 ≤ θ < 2π

Step 1: Rewrite secθ as 1/cosθ. - 1/cosθ = 2 → cosθ = 1/2 Step 2: Solve cosθ = 1/2 in the given range. - θ = π/3, 5π/3 (60°, 300°) What we did and why: We used the reciprocal identity to convert secθ to cosθ, then solved the basic equation.

Example 2 – Medium: Simplify sin(θ + π/6) + sin(θ – π/6)

Step 1: Use the compound angle formula for sin(A ± B). - sin(θ + π/6) = sinθ cos(π/6) + cosθ sin(π/6) - sin(θ – π/6) = sinθ cos(π/6) – cosθ sin(π/6) Step 2: Add them: - sinθ cos(π/6) + cosθ sin(π/6) + sinθ cos(π/6) – cosθ sin(π/6) = 2 sinθ cos(π/6) Step 3: Substitute exact values: - cos(π/6) = √3/2 → 2 sinθ (√3/2) = √3 sinθ What we did and why: We expanded using compound angles, then simplified by combining like terms. This is a common exam question—always expand first!

Example 3 – Exam-Style: Solve 3sinθ + 4cosθ = 2 for 0 ≤ θ < 2π

Step 1: Rewrite in R-form. - R = √(3² + 4²) = 5 - tanα = 4/3 → α = 0.927 radians (53.1°) - So, 3sinθ + 4cosθ = 5sin(θ + 0.927) Step 2: Set equal to 2: - 5sin(θ + 0.927) = 2 → sin(θ + 0.927) = 0.4 Step 3: Solve sin(x) = 0.4 (where x = θ + 0.927). - x = 0.412, π – 0.412 = 2.730 Step 4: Subtract α to find θ: - θ + 0.927 = 0.412 → θ = -0.515 (not in range) - θ + 0.927 = 2.730 → θ = 1.803 - Also, add 2π to the first solution: θ + 0.927 = 0.412 + 2π → θ = 5.769 Step 5: Check range: θ = 1.803, 5.769 (radians) What we did and why: We used R-form to convert the equation into a single sine function, making it easier to solve. This is a must-know technique for A-Level exams.

Common Mistakes

Exam Traps

1-Minute Recap

"Right, listen up—this is your last-minute checklist for trig identities. First, memorise the key formulas: reciprocal identities (sec = 1/cos, etc.), compound angles (sin(A±B)), and double angles (sin2θ = 2sinθcosθ). For R-form, always find R first (√(a² + b²)), then the phase shift (tanα = b/a). If the question says ‘θ is small,’ switch to radians immediately and use sinθ ≈ θ. When proving identities, start with the messier side and simplify. For equations, always check the range—don’t lose marks by missing solutions. And if you see sec, cosec, or cot, rewrite them in terms of sin/cos straight away. You’ve got this—go smash that exam!