University:
York University
Course:
MATH 1013 | Applied Calculus
Academic year:

2023
Views:

319

Pages:

2
Author:

Niklas

II. Trigonometric Expressions (Ch. 4 & 5) I. FORMULAS: Sum/difference formulas: sin(α + β) = sinα cosβ + cosα sinβ sin(α - β) = sinα cosβ - cosα sinβ cos(α + β) = cosα cosβ - sinα sinβ cos(α - β) = cosα cosβ + sinα sinβ tan(α + β) = (tanα + tanβ) / (1 - tanα tanβ) tan(α - β) = (tanα - tanβ) / (1 + tanα tanβ) Double-angle formulas: sin 2θ = 2sinθ cosθ cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ tan 2θ = (2tanθ) / (1 - tan²θ) Half-angle formulas: sin(θ/2) = ±√((1 - cosθ) / 2) cos(θ/2) = ±√((1 + cosθ) / 2) tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ)) (choose + or - based on quadrant) II. EVALUATE USING SUM/DIFFERENCE: cos(45 - 30) = cos45 cos30 + sin45 sin30 = (√6 + √2) / 4 tan(45 + 30) = (tan45 + tan30) / (1 - tan45 tan30) = (3 + √3) / (3 - √3) = 2 + √3 sin(105) = sin(60 + 45) = sin60 cos45 + cos60 sin45 = (√6 + √2) / 4 condense cos72cos130 + sin72sin130 = cos(72 - 130) = cos(-58) = cos58 sin(120) = sin(2(60)) = 2sin60cos60 = 2(√3/2)(1/2) = √3/2 tan(60) = tan(2(30)) = (2tan30) / (1-tan^2(30)) = (2√3/3) / (1-2/3) = √3 cos(90) = cos(2(45)) = 1-2sin^2(45) = 1-2(1/2)^2 = 0 IV. EVALUATE USING HALF-ANGLE: tan(45) = sqrt((1-cos(90))/(1+cos(90))) = 1 cos(30) = sqrt((1+cos(60))/2) = sqrt((1+1/2)/2) = √3/2 V. INVERSE TRIGONOMETRIC FUNCTIONS: let dom(f(x)) = D, VxED, f(f^-1(x)) = f^-1(f(x)) = x VI. EVALUATE: sin(sin^-1(√2/2)) = √2/2 tan(sin^-1(-1/2)) = tan(-π/6) = -√3/3 sin(cos^-1(√2/2)) = sin(π/4) = √2/2