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How to Create Multiple Sine Waves With Phase Shift

A sine wave is a graphical, mathematical description of a repetitive oscillation. They appear in pure mathematics and in many scientific fields, such as physics and electrical engineering. The general equation of a sine wave is "f(x) = a*sin(bx + c) + d" where "a" is the amplitude of the wave, "b" is the "stretch" of the wave, and "c" and "d" are the horizontal and vertical "shifts" of the wave. Sine wave phase shifts can occur when adding multiple waves together or in isolation.

Instructions

- 1
  Begin with the general trigonometric sine function "f(x) = a*sin(bx + c) + d" where a, b, c and d are known constants and x is a variable.
- 2
  Add a constant value to the x value within the sine function. For example, "f(x) = a*sin(bx + c) + d = sin(x + c)" when "a = 1," "b = 1" and "d = 0." Adding "c = 6" to the x value within the function phase shifts the sine function "6" units to the left and the sine equation becomes: "f(x) = sin(x + 6)."
- 3
  Subtract a constant value from the x value within the sine function. For example, "f(x) = a*sin(bx + (-c)) + d = sin(x + (-c))" when "a = 1," "b = 1" and "d = 0." Subtracting "c = 6" from the x value within the function phase shifts the sine function "6" units to the right and the sine equation becomes: "f(x) = sin(x + (-6)) = sin(x - 6)."
- 4
  Multiply the x variable by a constant to compress or stretch the graph of the sine function, depending on the sign of the constant. For example, the sine function "f(x) = sin(2x + 6)" stretches the function by a factor of "2" and phase shifts it to the left "6" units, thereby increasing the function's period. The function "f(x) = sin(-2x + 6)" compresses the function by a factor of "2" and phase shifts it to the right "6" units.

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