Topic 3 covers trigonometric functions, identities, and vector geometry. SL includes the unit circle, trig identities, sine/cosine rules, and basic vectors. HL extends to vector equations of lines/planes, cross products, and advanced trig.
Radian measure: π rad = 180°. The unit circle defines sin θ, cos θ, and tan θ for all angles. Arc length s = rθ; sector area A = ½r²θ. Exact values for 0°, 30°, 45°, 60°, 90° and their radian equivalents. Graphs of sin, cos, tan with amplitude, period, and phase shift.
Pythagorean: sin²θ + cos²θ = 1. Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ. Compound angle: sin(A ± B) = sinA cosB ± cosA sinB. These allow simplification and solving of trig equations. HL adds: t-formulas, sum-to-product, and proof of identities.
Sine rule: a/sinA = b/sinB = c/sinC (use for AAS, ASA). Cosine rule: c² = a² + b² − 2ab·cosC (use for SAS, SSS). Area = ½ab·sinC. The ambiguous case (SSA) in the sine rule where two triangles may be possible.
SL: vectors in 2D and 3D, magnitude, unit vectors, scalar (dot) product a·b = |a||b|cos θ. Parallel vectors: a = kb. Perpendicular vectors: a·b = 0. HL adds: vector equation of a line r = a + tb, angle between two lines, vector (cross) product a×b, vector equation of a plane, distance from a point to a line/plane.
Use the sine rule when you know an angle and its opposite side plus one more element (AAS, ASA, or SSA). Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS). The cosine rule is also the safer choice when the sine rule might give an ambiguous result.
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