University:
Carleton College
Course:
MATH 111 | Introduction to Calculus
Academic year:

2024
Views:

369

Pages:

6
Author:

SpaceCadet101

7.1-7.3 Quiz Review Name: Pre-Calculus Hour: Simplify the trigonometric expression. tan x csc x = sin x / cos x * 1 / sin x = 1 / cos x = sec x cos²x(1 + tan²x) = cos²x(sec²x) = cos²x * 1/cos²x = 1 Verify the identity. (cos x/sin x)(sin x/cos x) = 1 csc x(csc x - sin x) = cot²x = csc²x - 1 = 1 + cot²x - 1 = cot²x (sin x + cos x)² = 1 + 2 sin x cos x (sin x + cos x)(sin x + cos x) = sin²x + sin x cos x + sin x cos x + cos²x = 1 + 2 sin x cos x csc x - sin x = cot x cos x 1/sin x - sin x = (1 - sin²x) / sin x = cos²x / sin x = cos x * cos x / sin x = cot x cos x (cos x / sec x) - (csc x / sin x) = csc x - sin x cos²x - 1 / sin x = csc x - sin x (cos²x - 1) / sin x = csc x - sin x (1 - sin²x) / sin x = csc x - sin x 1/sin x - sin²x / sin x = csc x - sin x cot²x cos²x = cot²x - cos²x cos²x / sin²x - cos²x = (cos²x - cos²x sin²x) / sin²x = cos²x(1 - sin²x) / sin²x = cos²x * cos²x / sin²x = cot²x cos²x 9. (1 - cos x)/sin x - cos x/sin x = csc x (1 - cos x - cos x)/sin x = csc x (1 - 2cos x)/sin x = csc x (sin²x + cos²x - 2cos x)/sin x (1 - cos x) = csc x (sin x - cos x)(sin x - cos x)/sin x (1 - cos x) = csc x (sin x - cos x)/sin x = csc x 10. sin x(1/sin x - cos x/cos x) = cos²x sin x(1/sin x - 1) = cos²x 1 - sin²x = cos²x cos²x = cos²x 11. sin x cos x tan x + cos²x = 1 sin x cos x * sin x/cos x + cos²x = 1 sin²x + cos²x = 1 1 = 1 12. sin(π/2 - x) - sin(π/2 + x) = 0 sin(π/2)cos x - cos(π/2)sin x - (sin(π/2)cos x + cos(π/2)sin x) = 0 (1)cos x - (0)sin x - (1)cos x - (0)sin x = 0 cos x - 0 - cos x - 0 = 0 0 = 0 13. sin x/cos x + cos x/sin x = tan x + cot x (sin²x + cos²x)/cos x sin x = tan x + cot x 1/cos x sin x = tan x + cot x 14. cos x(cos x + 1) + sin x(sin x + 0) = -cos x cos²x + cos x + sin²x = -cos x 1 + cos x = -cos x 1 = -2cos x (1 - tan x tan y)/(cos x cos y) = 1 - tan x tan y (cos x cos y - sin x sin y)/(cos x cos y) = 1 - tan x tan y (cos x cos y)/(cos x cos y) - (sin x sin y)/(cos x cos y) = 1 - tan x tan y 1 - tan x tan y = 1 - tan x tan y 16. cos x - sin x = (cos²x + sin²x)/(cos x + sin x) cos x - sin x = (cos x + sin x)(cos x - sin x)/(cos x + sin x) cos x - sin x = cos x - sin x 17. (2 tan x)/(1 + tan²x) = 2 sin x cos x (2 sin x/cos x)/(1 + sin²x/cos²x) = 2 sin x cos x (2 sin x/cos x)/(1 + sin²x/cos²x) = 2 sin x cos x (2 sin x/cos x)/(cos²x + sin²x)/cos²x = 2 sin x cos x 2 sin x/cos x * cos²x/1 = 2 sin x cos x 2 sin x cos x = 2 sin x cos x 18. cos x - 1 = (cos 2x - 1)/(2(cos x + 1)) cos x - 1 = (2cos²x - 1 - 1)/(2(cos x + 1)) cos x - 1 = (2cos²x - 2)/(2(cos x + 1)) cos x - 1 = 2(cos²x - 1)/(2(cos x + 1)) cos x - 1 = (cos x + 1)(cos x - 1)/(cos x + 1) cos x - 1 = cos x - 1 19. √((1-cos x)/2) * √((1+cos x)/2) = sin x/2 √((1-cos x)(1+cos x)/4) = sin x/2 √((1-cos²x)/4) = sin x/2 √(sin²x/4) = sin x/2 sin x/2 = sin x/2 20. (sec x - 1)/(sin x sec x) = tan(x/2) (1/cos x - 1)/(sin x/cos x) = tan(x/2) (1 - cos x)/sin x = tan(x/2) (1 - cos x)/sin x = (1 - cos x)/(1 + cos x)