Compound Angles & R-MethodJEE Main

Concept

The compound-angle formulas expand trig functions of sums: sin(A+B) = sinAcosB + cosAsinB, cos(A+B) = cosAcosB − sinAsinB. Their killer application is the R-method: any $a\sin x + b\cos x$ collapses to a single sinusoid $R\sin(x + \varphi)$ with $R = \sqrt{a^2+b^2}$ — instantly giving the max, min and period.

Key formula

Derivation

Compound formulas come from the unit circle (rotation composition) or Euler's $e^{i(A+B)} = e^{iA}e^{iB}$ split into parts.

R-method: write a = Rcosφ, b = Rsinφ (possible since (a/R)² + (b/R)² = 1); then a·sinx + b·cosx = R(sinx·cosφ + cosx·sinφ) = R·sin(x+φ).

Scenarios to explore

• Compound Angles & R-Method — sin(A+B) proofs and the a·sinx + b·cosx collapse to one wave.

Real-world applications

• Max/min of 3sinx + 4cosx = ±5 — one line, no calculus.
• Superposing same-frequency waves (physics interference).
• Solving a·sinx + b·cosx = c: solvable ⟺ |c| ≤ R.

JEE exam tips

• tan(A+B) = (tanA + tanB)/(1 − tanAtanB); if A + B = 45°, then (1+tanA)(1+tanB) = 2.
• Range of a·sinx + b·cosx + c is [c − R, c + R].
• sin(A+B)·sin(A−B) = sin²A − sin²B — the difference-of-squares gem.

Common mistakes

• sin(A+B) = sinA + sinB — the classic disaster.
• Wrong sign in cos(A+B) (it's MINUS sinAsinB).
• Quadrant of φ from tanφ = b/a alone — check signs of a and b.

Exam traps to avoid

• a·sinx + b·cosx = c has NO solution when c² > a² + b² — check before solving.
• cos(A−B) − cos(A+B) = 2sinAsinB — product-to-sum pairs are these formulas re-shuffled.