Trigonometry extends from IGCSE into radians, identities, and equation-solving. These tools are essential for calculus, mechanics, and further pure mathematics.
Radians: π rad = 180°. Convert: multiply by π/180 (degrees to radians) or 180/π (radians to degrees). Arc length s = rθ. Sector area A = ½r²θ (θ in radians). Segment area = sector area − triangle area = ½r²(θ − sinθ). Use radians for all calculus work.
Fundamental identity: sin²θ + cos²θ ≡ 1. Also: tanθ ≡ sinθ/cosθ. Exact values: sin30° = ½, cos30° = √3/2, tan30° = 1/√3, sin45° = √2/2, sin60° = √3/2. CAST diagram for signs in each quadrant. Compound angles (P3): sin(A±B) = sinAcosB ± cosAsinB, cos(A±B) = cosAcosB ∓ sinAsinB. Double angle: sin2A = 2sinAcosA, cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A.
Solve in given range — find principal value, then use symmetry/periodicity. For sinθ = k: θ and (180°−θ), then add multiples of 360°. For cosθ = k: θ and (360°−θ). For tanθ = k: θ and (θ+180°). Use identities to reduce to single trig function. Example: 2sin²x + sinx − 1 = 0 → quadratic in sinx. R-method (P3): asinθ + bcosθ = Rsin(θ+α) where R = √(a²+b²) and tanα = b/a — useful for maximum/minimum and solving equations.
Radians are the natural unit for angles in mathematics because they make calculus formulae clean and simple. In radians: d/dx(sinx) = cosx. In degrees: d/dx(sin x°) = (π/180)cos x° — an ugly constant appears. Similarly, arc length = rθ and sector area = ½r²θ only work directly in radians. The definition of a radian (arc length = radius) connects angle to length naturally. All advanced mathematics, physics, and engineering use radians as the standard angle measure.
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