University:
University of North Georgia
Course:
MATH 1113 | Precalculus
Academic year:

2024
Views:

377

Pages:

13
Author:

larbiwxcvbn

Honors Pre-Calculus - 2nd Semester Final Exam Review Trigonometry Solve for θ, 0 ≤ θ ≤ 2π, without using a calculator: sinθ = 1/2 θ = π/6, 5π/6 cscθ = √2 sinθ = 1/√2 = √2/2 θ = π/4, 3π/4 tanθ = -1 θ = 3π/4, 7π/4 cosθ = -2 -1 ≤ cosθ ≤ 1 No solution Solve for θ in degrees, giving all solutions. sinθ + 1 = 0 sinθ = -1 θ = 270° + 360°n cosθ + √3/2 = 0 cosθ = -√3/2 θ = 150° + 360°n or 210° + 360°n tanθ = undefined θ = 90° + 180°n Give the exact value of each expression. tan(π/3) = √3 cot(sin^(-1)(1/2)) sinθ = 1/2 θ = 30° cot(30°) = √3 sec(cos^(-1)(-3/5)) cosθ = -3/5 θ = 218° sec(218°) = -5/3 Given: y = (1/2)sin(x/2) 12. Given: y = 4 + 2cos(3x - π/2) a. Amplitude: 2 b. Period: 2π/3 c. Phase Shift: π/6 RIGHT d. Vertical Shift: 4 DOWN e. Graph at least two periods of the function [Image of the graph of y = 4 + 2cos(3x - π/2)] 13. Simplify: (1 - sinθ)(1 + cscθ)sinθ (1 - sinθ)(1 + 1/sinθ)sinθ sinθ(1 + 1/sinθ - sinθ - 1) 1 - sin²θ = cos²θ 14. Prove: secθ + cscθ = 1 + tanθ + cotθ (1/cosθ) + (1/sinθ) = 1 + sinθ/cosθ + cosθ/sinθ (sinθ + cosθ)/(cosθsinθ) = (cosθsinθ + sin²θ + cos²θ)/(cosθsinθ) (sinθ + cosθ)/(cosθsinθ) = 1/(cosθsinθ) secθ + cscθ = secθcscθ 15. Solve: 2cos²θ = 1 + sinθ for 0 ≤ θ ≤ 2π 2(1 - sin²θ) = 1 + sinθ 2 - 2sin²θ = 1 + sinθ 0 = 2sin²θ + sinθ - 1 0 = (2sinθ - 1)(sinθ + 1) sinθ = 1/2 or sinθ = -1 θ = π/6, 5π/6, 3π/2 16. Solve for x: √2cos(x) - 1 = 0, 0 ≤ x ≤ 2π cos(x) = 1/√2 = √2/2 x = π/4, 7π/4 17. Solve a right triangle ABC, if a = 6 and ∠A = 30° (Give exact lengths) b = 6√3 c = 12 18. A boy flying a kite is standing 20 ft from a point directly under the kite. If the string to the kite is 40 ft long, find the angle of elevation of the kite. cosθ = 20/40 θ = 60° 19. An airplane is at an elevation of 30,000 ft and approaches the airport, with an angle of descent of 5°. What is the distance between the airport and the point on the ground directly below the airplane? tan 85° = x/30000 x = 342901.6 ft