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

2020
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61

Pages:

17
Author:

SpaceCadet101

Semester 2 Final Exam REVIEW Name: Key Pre-Calculus Chapter 7 - Analytic Trigonometry Simplify the expression. 1. cscθ - sinθ cscθ - sinθ * (cscθ + sinθ) / (cscθ + sinθ) (csc²θ - sin²θ) / (cscθ + sinθ) (1/sin²θ - sin²θ) / (1/sinθ + sinθ) (1 - sin⁴θ) / (1 + sin²θ) (1 - sin²θ)(1 + sin²θ) / (1 + sin²θ) 1 - sin²θ cos²θ 2. cosθ/(1 - sinθ) + (1 - sinθ)/cosθ (cos²θ + (1 - sinθ)²) / (cosθ(1 - sinθ)) (cos²θ + 1 - 2sinθ + sin²θ) / (cosθ(1 - sinθ)) (2 - 2sinθ) / (cosθ(1 - sinθ)) 2(1 - sinθ) / (cosθ(1 - sinθ)) 2/cosθ = 2secθ 3. sinθ + cosθ cotθ sinθ + cosθ * cosθ/sinθ (sin²θ + cos²θ) / sinθ 1/sinθ cscθ 4. sin⁴α + cos⁴α + 2sin²αcos²α (sin²α + cos²α)² 1² 1 5. sin(θ + φ) / sin(θ - φ) (sinθcosφ + cosθsinφ) / (sinθcosφ - cosθsinφ) Use an addition or subtraction formula to find the EXACT value of the expression. 11. sin 15° = sin(45° - 30°) = sin45°cos30° - cos45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4 12. cos 195° = cos(150° + 45°) = cos150°cos45° - sin150°sin45° = (-√3/2)(√2/2) - (1/2)(√2/2) = (-√6 - √2)/4 13. tan 105° = tan(60° + 45°) = (tan60° + tan45°) / (1 - tan60°tan45°) = (√3 + 1) / (1 - √3 * 1) = (√3 + 1) / (1 - √3) Find sin 2x, cos 2x, and tan 2x from the given information. 14. cos x = 4/5, x in quadrant I sin x = 3/5 sin 2x = 2sin x cos x = 2(3/5)(4/5) = 24/25 cos 2x = 2cos²x - 1 = 2(16/25) - 1 = 7/25 tan 2x = sin 2x / cos 2x = (24/25) / (7/25) = 24/7 15. csc x = 4, x in quadrant II sin x = 1/4, cos x = -√15/4 sin 2x = 2(1/4)(-√15/4) = -√15/8 cos 2x = 2(-√15/4)² - 1 = 7/8 tan 2x = (-√15/8) / (7/8) = -√15/7 Use a half-angle formula to find the EXACT value of the expression. 16. tan 22.5° tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ)) tan 22.5° = ±√((1 - cos45°) / (1 + cos45°)) = ±√((1 - √2/2) / (1 + √2/2)) = ±√((2 - √2) / (2 + √2)) = ±√((2 - √2)² / (4 - 2)) = ±(2 - √2)/√2 = ±(√2 - 1) 17. cos 15° cos(θ/2) = ±√((1 + cosθ) / 2) cos 15° = ±√((1 + cos30°) / 2) = ±√((1 + √3/2) / 2) = ±√((2 + √3) / 4) = ±√((2 + √3)/2) Find the EXACT value of each expression, if it is defined. 18. sin⁻¹(1/2) θ = 30° = π/6 19. cos⁻¹(-√3/2) θ = 150° = 5π/6 20. tan⁻¹(0) θ = 0° = 0 21. tan(sin⁻¹(4/5)) sinθ = 4/5 cosθ = 3/5 tanθ = 4/3 22. sin(tan⁻¹(12/5)) tanθ = 12/5 sinθ = 12/13 23. csc(cos⁻¹(7/25)) cosθ = 7/25 sinθ = 24/25 cscθ = 25/24 CHAPTER 9 - Systems, Matrices & Inequalities Solve the system. 1. y = x² + 8x y - 16 = 2x x² + 8x = 2x + 16 x² + 6x - 16 = 0 (x + 8)(x - 2) = 0 x = -8, 2 y = (-8)² + 8(-8) = 0 y = 2² + 8(2) = 20 (-8, 0), (2, 20) 2. x - y = 4 xy = 12 x = y + 4 (y + 4)y = 12 y² + 4y - 12 = 0 (y + 6)(y - 2) = 0 y = -6, 2 x = -6 + 4 = -2 x = 2 + 4 = 6 (-2, -6), (6, 2) The matrices A, B, C, D, and E are defined as follows. Carry out the operation if possible. A = [4 6; 1 3] B = [2 5; -3 7] C = [2 3; 1 0; 0 2] D = [10 6; -3 5; 2 -1] E = [1 -2 4; 3 7 2; 0 9 -1] 3. A + B [4 6; 1 3] + [2 5; -3 7] = [6 11; -2 10] 4. 3C - D [6 9; 3 0; 0 6] - [10 6; -3 5; 2 -1] = [-4 3; 6 -5; -2 7] 5. DA 3x2 2x2 [10 6; -3 5; 2 -1] * [4 6; 1 3] = [40+6 60+18; -12+5 -18+15; 8-1 12-3] = [46 78; -7 -3; 7 9] 6. |B| Determinant [2 5; -3 7] = 2(7) - (5)(-3) = 14 + 15 = 29

Pre-Calc Final Exam Review Key

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