Untitled
PART 1 — Degrees & Radians
1. Convert 60∘60^\circ to radians.
A. π6\frac{\pi}{6}
B. π3\frac{\pi}{3}
C. 2π3\frac{2\pi}{3}
D. 3π2\frac{3\pi}{2}
2. Convert 225∘225^\circ to radians.
A. 5π4\frac{5\pi}{4}
B. 3π4\frac{3\pi}{4}
C. 7π4\frac{7\pi}{4}
D. 9π4\frac{9\pi}{4}
3. Convert 5π6\frac{5\pi}{6} to degrees.
A. 120∘120^\circ
B. 135∘135^\circ
C. 150∘150^\circ
D. 180∘180^\circ
4. Convert −7π4-\frac{7\pi}{4} to degrees.
A. −225∘-225^\circ
B. −315∘-315^\circ
C. −360∘-360^\circ
D. −270∘-270^\circ
5. Convert 52∘22′2′′52^\circ22'2'' to decimal degrees.
A. 52.36∘52.36^\circ
B. 52.37∘52.37^\circ
C. 52.39∘52.39^\circ
D. 53.36∘53.36^\circ
6. Convert 39.65∘39.65^\circ to degrees, minutes, and seconds.
A. 39∘36′0′′39^\circ36'0''
B. 39∘39′0′′39^\circ39'0''
C. 39∘39′30′′39^\circ39'30''
D. 39∘65′0′′39^\circ65'0''
PART 2 — Arc Length & Area
7. A circle has radius 8 cm and central angle 45∘45^\circ. Find the arc length.
A. 2π2\pi cm
B. 4π4\pi cm
C. 8π8\pi cm
D. 16π16\pi cm
8. A circle has radius 10 ft and central angle 120∘120^\circ. Find the sector area.
A. 50π3\frac{50\pi}{3} ft²
B. 100π3\frac{100\pi}{3} ft²
C. 200π3\frac{200\pi}{3} ft²
D. 100π100\pi ft²
9. A sector has area 18 cm218\text{ cm}^2 and radius 5 cm. Find the central angle in radians.
A. 0.720.72
B. 1.441.44
C. 2.252.25
D. 3.603.60
10. A sprinkler sprays over a distance of 20 ft through an angle of 155∘155^\circ. Approximately how much lawn is covered?
A. 341.05 ft2341.05\text{ ft}^2
B. 441.05 ft2441.05\text{ ft}^2
C. 541.05 ft2541.05\text{ ft}^2
D. 641.05 ft2641.05\text{ ft}^2
PART 3 — Linear & Angular Speed
11. An object travels around a circle with radius 12 cm. In 20 seconds it sweeps through 13\frac13 radian. What is its linear speed?
A. 15\frac15 cm/s
B. 13\frac13 cm/s
C. 35\frac35 cm/s
D. 55 cm/s
12. Using the same information, what is the angular speed?
A. 120\frac1{20} rad/s
B. 130\frac1{30} rad/s
C. 160\frac1{60} rad/s
D. 320\frac3{20} rad/s
PART 4 — Six Trig Functions
13. A point on the terminal side of θ\theta is (−5,12)(-5,12). What is sinθ\sin\theta?
A. 513\frac5{13}
B. −513-\frac5{13}
C. 1213\frac{12}{13}
D. −1213-\frac{12}{13}
14. If sinθ=53\sin\theta=\frac{\sqrt5}{3} and θ\theta is in Quadrant II, what is cosθ\cos\theta?
A. 23\frac23
B. −23-\frac23
C. 53\frac{\sqrt5}{3}
D. −53-\frac{\sqrt5}{3}
15. If sinθ=115\sin\theta=\frac1{15}, find cscθ\csc\theta.
A. 115\frac1{15}
B. −15-15
C. 1515
D. 1414
16. If secθ=9\sec\theta=9 and cotθ>0\cot\theta>0, what is sinθ\sin\theta?
A. 49\frac49
B. 459\frac{4\sqrt5}{9}
C. 59\frac{\sqrt5}{9}
D. 19\frac19
PART 5 — Quadrants
17. If sinθ>0\sin\theta>0 and cosθ<0\cos\theta<0, which quadrant?
A. I
B. II
C. III
D. IV
18. If cscθ>0\csc\theta>0 and cosθ<0\cos\theta<0, which quadrant?
A. I
B. II
C. III
D. IV
19. In which quadrants is tangent positive?
A. I and II
B. II and III
C. I and III
D. III and IV
20. In which quadrants is cosine negative?
A. I and II
B. II and III
C. III and IV
D. I and IV
PART 6 — Unit Circle
21. Find:
sin2π3\sin\frac{2\pi}{3}
A. 12\frac12
B. −12-\frac12
C. 32\frac{\sqrt3}{2}
D. −32-\frac{\sqrt3}{2}
22. Find:
cosπ6\cos\frac{\pi}{6}
A. 12\frac12
B. 22\frac{\sqrt2}{2}
C. 32\frac{\sqrt3}{2}
D. −32-\frac{\sqrt3}{2}
23. Find:
tanπ4\tan\frac{\pi}{4}
A. 0
B. 12\frac12
C. 1
D. 3\sqrt3
24. Find:
secπ3\sec\frac{\pi}{3}
A. 12\frac12
B. 1
C. 2
D. 3\sqrt3
25. Find:
cscπ6\csc\frac{\pi}{6}
A. 12\frac12
B. 1
C. 2
D. 3\sqrt3
26. Find:
cotπ4\cot\frac{\pi}{4}
A. 0
B. 1
C. 2\sqrt2
D. 3\sqrt3
PART 7 — Identities
27. Simplify:
sin2(27∘)+cos2(27∘)\sin^2(27^\circ)+\cos^2(27^\circ)
A. 0
B. 12\frac12
C. 1
D. 2
28. Simplify:
sin48∘csc48∘\sin48^\circ\csc48^\circ
A. 0
B. 1
C. sin48∘\sin48^\circ
D. csc48∘\csc48^\circ
29. If sinθ=0.7\sin\theta=0.7, find:
sin(θ+2π)+sin(θ+4π)+sin(θ+6π)\sin(\theta+2\pi)+\sin(\theta+4\pi)+\sin(\theta+6\pi)
A. 0.7
B. 1.4
C. 2.1
D. 3.1
30. Complete:
tanθ⋅x=1\tan\theta\cdot\boxed{\phantom{x}}=1
A. tanθ\tan\theta
B. sinθ\sin\theta
C. cotθ\cot\theta
D. secθ\sec\theta
31. Complete:
cosθ⋅x=1\cos\theta\cdot\boxed{\phantom{x}}=1
A. sinθ\sin\theta
B. secθ\sec\theta
C. cscθ\csc\theta
D. cotθ\cot\theta
PART 8 — Domain & Range
32. What is the domain of y=sinxy=\sin x?
A. [−1,1][-1,1]
B. (−∞,∞)(-\infty,\infty)
C. x≠0x\neq0
D. [0,2π][0,2\pi]
33. What is the range of y=sinxy=\sin x?
A. (−∞,∞)(-\infty,\infty)
B. [0,1][0,1]
C. [−1,1][-1,1]
D. x≠0x\neq0
34. What is the domain of y=cscxy=\csc x?
A. All real numbers
B. x≠nπx\neq n\pi
C. x≠π2+nπx\neq\frac{\pi}{2}+n\pi
D. −1≤x≤1-1\le x\le1
35. What is the domain of y=tanxy=\tan x?
A. x≠nπx\neq n\pi
B. All real numbers
C. x≠π2+nπx\neq\frac{\pi}{2}+n\pi
D. −1≤x≤1-1\le x\le1
PART 9 — Amplitude & Period
36. Find the amplitude and period:
y=7cos(2x)y=7\cos(2x)
A. Amplitude 7, Period π\pi
B. Amplitude 2, Period 7π7\pi
C. Amplitude 7, Period 2π2\pi
D. Amplitude 14, Period π\pi
37. Find the amplitude and period:
y=−3cos(14x)y=-3\cos\left(\frac14x\right)
A. 3 and 2π2\pi
B. 3 and 8π8\pi
C. -3 and 8π8\pi
D. 4 and 6π6\pi
38. Find the amplitude and period:
y=−2sin(14x)y=-2\sin\left(\frac14x\right)
A. 2 and 8π8\pi
B. -2 and 8π8\pi
C. 2 and 4π4\pi
D. 4 and 2π2\pi
39. Find the amplitude and period:
y=5sin(3x)y=5\sin(3x)
A. 5 and 6π6\pi
B. 3 and 5π5\pi
C. 5 and 2π3\frac{2\pi}{3}
D. 3 and 2π5\frac{2\pi}{5}
PART 10 — Graphing
40. What are the 5 key points for one period of
y=3cosx?y=3\cos x?
A. (0,0),(π2,3),(π,0),(3π2,−3),(2π,0)(0,0),(\frac\pi2,3),(\pi,0),(\frac{3\pi}2,-3),(2\pi,0)
B. (0,3),(π2,0),(π,−3),(3π2,0),(2π,3)(0,3),(\frac\pi2,0),(\pi,-3),(\frac{3\pi}2,0),(2\pi,3)
C. (0,−3),(π2,0),(π,3),(3π2,0),(2π,−3)(0,-3),(\frac\pi2,0),(\pi,3),(\frac{3\pi}2,0),(2\pi,-3)
D. (0,3),(π,3),(2π,3),(3π,3),(4π,3)(0,3),(\pi,3),(2\pi,3),(3\pi,3),(4\pi,3)
41. For
y=3cos(π2x),y=3\cos\left(\frac{\pi}{2}x\right),
what is the period?
A. 2
B. 4
C. π\pi
D. 2π2\pi
42. For
y=−2sin(π4x),y=-2\sin\left(\frac{\pi}{4}x\right),
what is the period?
A. 2
B. 4
C. 8
D. 4π4\pi
PART 11 — Transformations
43. For
y=4sin(2x)−1,y=4\sin(2x)-1,
what is the vertical shift?
A. Up 1
B. Down 1
C. Up 4
D. Down 4
44. For
y=−3cosx+2,y=-3\cos x+2,
what is the maximum value?
A. -3
B. 2
C. 3
D. 5
45. For
y=2cos(4x+π2),y=2\cos\left(4x+\frac{\pi}{2}\right),
what is the phase shift?
A. Right π8\frac{\pi}{8}
B. Left π8\frac{\pi}{8}
C. Right π2\frac{\pi}{2}
D. Left π2\frac{\pi}{2}
46. For
y=7sin(12x+π8),y=7\sin\left(\frac12x+\frac{\pi}{8}\right),
what is the phase shift?
A. Right π4\frac{\pi}{4}
B. Left π4\frac{\pi}{4}
C. Right π8\frac{\pi}{8}
D. Left π8\frac{\pi}{8}
PART 12 — Write the Equation
47. A cosine graph has maximum 99, minimum −9-9, starts at a minimum at x=0x=0, and the next minimum occurs at x=4πx=4\pi. Which equation is correct?
A. y=9cosx2y=9\cos\frac{x}{2}
B. y=−9cosx2y=-9\cos\frac{x}{2}
C. y=9cos(2x)y=9\cos(2x)
D. y=−9cos(2x)y=-9\cos(2x)
48. A sine graph has amplitude 7, period 4π4\pi, and crosses the midline going upward at x=0x=0. Which equation?
A. y=7cosx2y=7\cos\frac{x}{2}
B. y=−7sinx2y=-7\sin\frac{x}{2}
C. y=7sinx2y=7\sin\frac{x}{2}
D. y=7sin(2x)y=7\sin(2x)
49. A cosine function has amplitude 2, period π2\frac{\pi}{2}, and starts at its maximum when x=0x=0. Which equation?
A. y=2cos(2x)y=2\cos(2x)
B. y=2cos(4x)y=2\cos(4x)
C. y=4cos(2x)y=4\cos(2x)
D. y=2sin(4x)y=2\sin(4x)
🔥 PART 13 — Challenge
50. If
cosθ=−513\cos\theta=-\frac5{13}
and θ\theta is in Quadrant II, what is sinθ\sin\theta?
A. 1213\frac{12}{13}
B. −1213-\frac{12}{13}
C. 513\frac5{13}
D. −513-\frac5{13}
51. For
y=−4cos(2x−π3)+2,y=-4\cos\left(2x-\frac{\pi}{3}\right)+2,
what is the phase shift?
A. Left π6\frac{\pi}{6}
B. Right π6\frac{\pi}{6}
C. Left π3\frac{\pi}{3}
D. Right π3\frac{\pi}{3}
52. Write the equation of a cosine function with:
Amplitude = 5
Period = 2π2\pi
Shift right π4\frac{\pi}{4}
Vertical shift = -2
Reflected across the x-axis
A. y=5cos(x+π4)−2y=5\cos(x+\frac{\pi}{4})-2
B. y=−5cos(x+π4)−2y=-5\cos(x+\frac{\pi}{4})-2
C. y=−5cos(x−π4)−2y=-5\cos(x-\frac{\pi}{4})-2
D. y=5cos(x−4π)+2