day 2 graphing sine and cosine worksheet (graded)\n1. find all possible angles for θ using the domain ( 0…

day 2 graphing sine and cosine worksheet (graded)\n1. find all possible angles for θ using the domain ( 0 leq \theta < 2pi ).\n a) ( sin \theta = \frac{sqrt{3}}{2} ) b) ( cos \theta = -\frac{1}{2} ) c) ( cos \theta = 1 ) d) ( cos \theta = \frac{sqrt{2}}{2} )\n2. ( y = 4 sin ( 2 x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n3. ( y = 2 sin ( 4 x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n4. ( y = 2 + cos ( x - \frac { pi } { 4 } ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n5. ( y = - 3 - cos ( x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________

day 2 graphing sine and cosine worksheet (graded)\n1. find all possible angles for θ using the domain ( 0 leq \theta < 2pi ).\n a) ( sin \theta = \frac{sqrt{3}}{2} ) b) ( cos \theta = -\frac{1}{2} ) c) ( cos \theta = 1 ) d) ( cos \theta = \frac{sqrt{2}}{2} )\n2. ( y = 4 sin ( 2 x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n3. ( y = 2 sin ( 4 x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n4. ( y = 2 + cos ( x - \frac { pi } { 4 } ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________\n5. ( y = - 3 - cos ( x ) )\nphase shift: ________ period: ________\nvertical shift: ________ amplitude: ________

Answer

Explanation:

Step1: Recall the general form of a sinusoidal function

The general form of a sinusoidal function is (y = A\sin(B(x - C))+D) or (y = A\cos(B(x - C))+D), where:

  • Amplitude (|A|)
  • Period (T=\frac{2\pi}{|B|})
  • Phase - shift (C) (positive is to the right, negative is to the left)
  • Vertical - shift (D)

Step2: Solve for problem 2 ((y = 4\sin(2x)))

  • For (y = 4\sin(2x)), comparing with (y = A\sin(B(x - C))+D), we have (A = 4), (B = 2), (C = 0), (D = 0)
  • Amplitude: (|A|=4)
  • Period: (T=\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi)
  • Phase - shift: (C = 0)
  • Vertical - shift: (D = 0)

Step3: Solve for problem 3 ((y = 2\sin(4x)))

  • For (y = 2\sin(4x)), comparing with (y = A\sin(B(x - C))+D), we have (A = 2), (B = 4), (C = 0), (D = 0)
  • Amplitude: (|A|=2)
  • Period: (T=\frac{2\pi}{|B|}=\frac{2\pi}{4}=\frac{\pi}{2})
  • Phase - shift: (C = 0)
  • Vertical - shift: (D = 0)

Step4: Solve for problem 4 ((y = 2+\cos(x-\frac{\pi}{4})))

  • For (y = 2+\cos(x - \frac{\pi}{4})), comparing with (y=A\cos(B(x - C))+D), we have (A = 1), (B = 1), (C=\frac{\pi}{4}), (D = 2)
  • Amplitude: (|A| = 1)
  • Period: (T=\frac{2\pi}{|B|}=2\pi)
  • Phase - shift: (C=\frac{\pi}{4}) (shift to the right by (\frac{\pi}{4}))
  • Vertical - shift: (D = 2)

Step5: Solve for problem 5 ((y=-3-\cos(x)))

  • For (y=-3-\cos(x)), comparing with (y = A\cos(B(x - C))+D), we have (A=- 1), (B = 1), (C = 0), (D=-3)
  • Amplitude: (|A| = 1)
  • Period: (T=\frac{2\pi}{|B|}=2\pi)
  • Phase - shift: (C = 0)
  • Vertical - shift: (D=-3)

Answer:

  1. Phase shift: (0); Period: (\pi); Vertical shift: (0); Amplitude: (4)
  2. Phase shift: (0); Period: (\frac{\pi}{2}); Vertical shift: (0); Amplitude: (2)
  3. Phase shift: (\frac{\pi}{4}); Period: (2\pi); Vertical shift: (2); Amplitude: (1)
  4. Phase shift: (0); Period: (2\pi); Vertical shift: (-3); Amplitude: (1)