University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

447

Pages:

3
Author:

Luciano Kirby

Question 1 cot²x - 1 Express in terms of sin x: csc²x Answer (cot²(x) - 1) / csc²(x) = (cos²(x)/sin²(x) - 1) / (1/sin²(x)) = (cos²(x) - sin²(x)) / sin²(x) * sin²(x) = cos²(x) - sin²(x) = (1 - sin²(x)) - sin²(x) = 1 - 2sin²(x) Question 2 if cos 45 = 1/√2 find cos 315, sin 270, sin 210, tan 210 Answer cos 45 = 1/√2, sin 45 = √(1 - cos²(45)) = √(1 - (1/2)) = 1/√2 = cos 45, sin 90 = sin (2 * 45) = 2sin 45 cos 45 = 2 * (1/√2) * (1/√2) = 1, cos 90 = cos (2*45) = cos 45 * cos 45 - sin 45 * sin 45 = 1/2 - 1/2 = 0, sin 180 = sin (2 * 90) = 2sin 90 cos 90 = 0, cos 180 = cos (2 * 90) = cos 90 cos 90 - sin 90 sin 90 = 0 - 1 = -1, cos (X) = cos (-X), so cos 45 = cos(-45) = cos (360-45) = cos (315) = 1/√2. sin 270 = sin (180 + 90) = sin 180 cos 90 + cos 180 sin 90 = 0 - 1 = -1. sin 3X = sin (2X + X) = sin 2Xcos X + cos 2Xsin X = 2sin Xcos²(X) + (cos²(X) - sin²(X))sin(X) if 3X = 90: 1 = 2sin 30(1 - sin²(30)) + (1 - sin²(30))sin (30) - sin³(30) 2sin 30 - 2sin³(30) + sin (30) - sin³(30) - sin³(30) Question 3 Verify the identity sec x - sec x sin²x = cos x Answer sec x - sec x sin²x = cos x sec x (1 - sin²x) = cos x sec x cos²x = cos x (1 / cos x) * cos²x = cos x cos x = cos x Question 4 Solve sec(30) + tan(45) - (csc(60) / sec(30)) + cos(60) + cot(45) Answer cos ec 60 = 1 / sin 60 = 2/√3, sec 30 = 1 / cos 30 = 2/√3 cot 45 = tan 45 = 1 cos 60 = sin 30 = 1/2 The expression becomes: 2/√3 + 1 - (2/√3 / 2/√3) + 1/2 + 1 = 2/√3 + 1 - 1 + 1/2 + 1 = 2/√3 + 3/2 = 2/√3 + 3/2 ≈ 2.6547 Question 5 Simplify the expression (sec x / sin x) / (sin x / cos x) Answer (sec x / sin x) / (sin x / cos x) = (1 / cos x) / sin x * (cos x / sin x) = 1 / sin²x = cos²x / sin²x = (cos²x / sin²x) / 1 = cos²x / sin²x = (1 - sin²x) / sin²x = sin²x - 1 + sin²x / sin²x = 1