Compound angle identities allow us to evaluate trigonometric expressions involving the sum or difference of angles.
They are foundational tools for simplifying expressions, solving trigonometric equations, and proving relationships.
This topic also includes the double-angle identity for tan, which is derived from the compound-angle formula.
| Concept | Formula / Meaning |
|---|---|
| Compound angle identity (sin) | sin(A + B) = sinA × cosB + cosA × sinB sin(A − B) = sinA × cosB − cosA × sinB |
| Compound angle identity (cos) | cos(A + B) = cosA × cosB − sinA × sinB cos(A − B) = cosA × cosB + sinA × sinB |
| Compound angle identity (tan) | tan(A + B) = (tanA + tanB) ÷ (1 − tanA × tanB) tan(A − B) = (tanA − tanB) ÷ (1 + tanA × tanB) |
| Double-angle identity for tan | tan(2A) = (2 × tanA) ÷ (1 − tan2A) |
📌 Why compound angles are important
- Allow evaluation of trigonometric values for angles that are not on the standard unit-circle table.
- Enable simplification of expressions before differentiation or integration.
- Essential in solving equations involving multiple angles.
- Used extensively in physics (wave interference, AC circuits, oscillations).
📌 Worked example (concise, exam-style)
Example: Evaluate sin(75°) exactly.
Step 1 — Express 75° as a sum:
75° = 45° + 30°
Step 2 — Use the identity:
sin(45° + 30°) = sin45° × cos30° + cos45° × sin30°
Step 3 — Substitute exact values:
sin45° = √2 ÷ 2, cos45° = √2 ÷ 2
cos30° = √3 ÷ 2, sin30° = 1 ÷ 2
Step 4 — Compute:
sin75° = (√2 ÷ 2)(√3 ÷ 2) + (√2 ÷ 2)(1 ÷ 2)
sin75° = (√6 ÷ 4) + (√2 ÷ 4)
= (√6 + √2) ÷ 4
🌍 Real-World Applications
- AC electrical systems use compound angle identities to analyse phase shifts between voltage and current.
- GPS triangulation algorithms depend on compound angle relationships for accurate positioning.
- Wave interference patterns in physics rely heavily on angle-sum trigonometry.
📐 IA Spotlight
- Investigating how compound-angle identities appear in oscillations or resonance.
- Modelling two-source wave interference using sin(A ± B) expressions.
- Exploring GPS triangulation mathematics through angle sum relationships.
🔍 TOK Perspective
- Compound identities originate from geometric proofs. How does mathematical proof differ from scientific evidence?
- Why do different methods of proof (geometric, algebraic, analytic) all confirm the same identities?
- What does this reveal about the interconnected structure of mathematical knowledge?
📝 Exam Tips
- Always choose the simplest angle pair for evaluating expressions (e.g., 45° + 30°, not 60° + 15°).
- Keep exact values in radical form unless the question requests decimals.
- For tan(2A), check that tanA is defined—avoid values where tanA is undefined.