c. c. (0.503x = − + d. 22.61sin 2 .038) y = −1.814) + d. 25.693 sin(0.476 49.854 y x 57.17 e. 90 e. 80 0 13 20 0 13 31. a. 20 32. a. = 77.0 − 31.8 = 45.2 = A Amplitude: b. = 75.4 − 25.5 = 49.9 = 22.6 A b. Amplitude: 2 2 24.95 2 2 ω = = π 2π Vertical Shift: = = 75.4+25.5 100.9 50.45 Vertical Shift: 2 2 ω = = π 2π 77.0+31.8 = 108.8 = 54.4 12 6 2 2 Phase shift (use = 25.5, = y x 12 6 1): Phase shift (use = 31.8, = 1) : y x φ 25.5 = 24.95 sin ⎛ π ⋅1 − ⎞ + ⎛ π ⎞ 50.45 ⎜ 6 ⎟ φ ⋅1 − + 54.4 31.8 = 22.6 sin ⎝ ⎠ ⎜ 6 ⎟ ⎝ ⎠ − 24.95 = 24.95 sin ⎛ π − ⎛ π ⎞ φ ⎞ ⎜ ⎟ 6 − = − φ 22.6 22.6 sin ⎝ ⎠ ⎜ 6 ⎟ −1 = sin ⎛ π − ⎝ ⎠ φ ⎞ ⎛ π ⎞ ⎜ 6 ⎟ φ −1 = sin − ⎝ ⎠ ⎜ 6 ⎟ ⎝ ⎠ φ − π = π − π π = − 2 6 φ φ = 2π 2 6 2π − Thus,
c. d. y = 22.61sin (0.503x − 2 .038) + 57.17 c. d. y = 25.693 sin(0.476 x −1.814) + 49.854 e. 90 e. 80 0 13 20 31. a. 0 13 20 32. a. b. Amplitude: A = 75.4 − 25.5 = 49.9 = 24.95 2 2 b. Amplitude: A = 77.0 − 31.8 = 45.2 = 22.6 2 2 Vertical Shift: ω = 2π = π 12 6 75.4+25.5 = 100.9 = 50.45 2 2 Vertical Shift: ω = 2π = π 77.0+31.8 = 108.8 = 54.4 2 2 Phase shift (use y = 25.5, x = 1): 12 6 Phase shift (use y = 31.8, x = 1) : 25.5 = 24.95 sin ⎛ π ⋅1 − φ ⎞ + 50.45 ⎛ π ⎞ ⎜ 6 ⎟ 31.8 = 22.6 sin ⋅1 − φ + 54.4 ⎝ ⎠ ⎜ 6 ⎟ ⎝ ⎠ − 24.95 = 24.95 sin ⎛ π − φ ⎞ ⎛ π ⎞ ⎜ 6 ⎟ − 22.6 = 22.6 sin − φ ⎝ ⎠ ⎜ 6 ⎟ −1 = sin ⎛ π − φ ⎞ ⎝ ⎠ ⎛ π ⎞ ⎜ 6 ⎟ −1 = sin − φ ⎝ ⎠ ⎜ 6 ⎟ ⎝ ⎠ − π = π − φ π π 2 6 φ = 2π − = − φ 2 6 2π Thus,
62514-0132318806_ISM_02-1.docx