Advanced ACT Math: Logarithms, Complex Numbers, and Trigonometry
Advanced ACT Math questions (typically questions 40–60) test content beyond standard Algebra II in a way that rewards students who have mastered the concepts rather than just practiced the procedures. Logarithms: log_b(x) = y means b^y = x. Key log rules tested: log(ab) = log(a) + log(b); log(a/b) = log(a) − log(b); log(a^n) = n × log(a). The change of base formula: log_b(x) = ln(x)/ln(b) = log(x)/log(b). ACT log questions typically ask you to evaluate log expressions using rules, or to solve equations like log₂(x) = 3 → x = 2³ = 8. Complex numbers: a + bi where i = √(−1) and i² = −1. Operations: add/subtract like terms; multiply using FOIL and substitute i² = −1; divide by multiplying numerator and denominator by the conjugate (a − bi). The modulus |a + bi| = √(a² + b²). Trigonometry at ACT level: unit circle values (sin, cos, tan at 0°, 30°, 45°, 60°, 90°), reciprocal functions (csc = 1/sin, sec = 1/cos, cot = 1/tan), the Pythagorean identity sin²θ + cos²θ = 1, and the law of sines (a/sin A = b/sin B) and law of cosines (c² = a² + b² − 2ab cos C) for non-right triangles. Graphs of trigonometric functions: y = A sin(Bx + C) + D — amplitude is |A|, period is 2π/B, phase shift is −C/B, vertical shift is D. These parameters are tested in both algebraic and graphical forms.