Trigonometry in the IAL builds from basic ratios to advanced identities — double angles, harmonic form, reciprocal trig functions, and proofs — essential for calculus and mechanics.
Radians: π rad = 180°. Arc length s = rθ, sector area A = ½r²θ. Fundamental identities: sin²θ + cos²θ ≡ 1, tanθ ≡ sinθ/cosθ. Solving equations: find principal value, then use symmetry of graph or CAST diagram for all solutions in the required interval. Addition formulae: sin(A±B) = sinAcosB ± cosAsinB; cos(A±B) = cosAcosB ∓ sinAsinB; tan(A±B) = (tanA ± tanB)/(1 ∓ tanAtanB). Double angle: sin2A = 2sinAcosA; cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A; tan2A = 2tanA/(1 − tan²A).
Harmonic form: asinθ + bcosθ ≡ Rsin(θ + α) where R = √(a²+b²), tanα = b/a. Used to find max/min and solve equations. Reciprocal functions: secθ = 1/cosθ, cosecθ = 1/sinθ, cotθ = cosθ/sinθ. Identities: 1 + tan²θ ≡ sec²θ; 1 + cot²θ ≡ cosec²θ. Inverse functions: arcsin, arccos, arctan with restricted domains. Small angle approximations: sinθ ≈ θ, cosθ ≈ 1 − θ²/2, tanθ ≈ θ (θ in radians). Factor formulae: sinP + sinQ = 2sin((P+Q)/2)cos((P−Q)/2), etc.
Step-by-step: (1) Rearrange to get a single trig function: sinx = k, cosx = k, or tanx = k. If quadratic in sinx/cosx, factorise or use the quadratic formula. (2) Find the principal value using inverse trig (calculator gives one solution). (3) Use the CAST diagram or graph to find all solutions in the required interval. For sinx = k: solutions are x and (180° − x) in one cycle. For cosx = k: solutions are x and (360° − x). For tanx = k: solutions repeat every 180°. (4) If the argument is modified (e.g., sin(2x+30°) = k), first solve for the bracket, then solve for x — making sure to adjust the interval for the bracket (e.g., if 0 ≤ x ≤ 360°, then 30° ≤ 2x+30° ≤ 750°).
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