Sine and Cosine Wave Flashcards

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Standardized tests such as the ACT will not get much more difficult than simply matching a graph to an equation. The equations for sine and cosine wave functions have two coefficients, A and B, and are expressed this way:

A{sin (Bθ)}

The coefficient A governs the amplitude of the graph (how tall it is). The coefficient B governs the period of the graph ( how long it takes to complete a cycle).

The equation for the graph below is expressed simply as y = sin θ or even simply y = sin x. The coefficients A and B are both 1.

The equation could be written as y = 1(sin)1(θ) or y = 1(sin)1(x). You can see that the maximum height of the graph is 1 and the minimum is -1; therefore, the amplitude is 1. You can also see that the graph completes a cycle in 2π, so its period is also 1.

You can see that the graph below begins at 1 on the y axis, which indicates a cosine wave. The amplitude is 1, and the period is 1.

When the A coefficient increases, the amplitude increases directly.

When the B coefficient increases, the period decreases inversely. So, if B=2, then the period is cut in half. If B=1/2, the period is doubled.

One More Graph Before We Try Flashcards

See whether you can express the equation for the graph below.

First, determine whether it is sine or cosine. It begins at y=0, so it is a sine wave. Next, determine the amplitude. The wave reaches up to 2 and down to -2, so the amplitude is 2.
Next, determine the period. The wave completes a cycle in 4π, so the period is 1/2. Therefore, this graph can be expressed with any of these equations:

y = 2 sin 1/2(θ)

y = 2 sin 1/2(x)

y = 2 sin x/2

Flashcard Practice


  • Is this wave sine or cosine?
    cosine
  • What is the amplitude?
    2
  • What is the period?
    1/2
  • What is the equation for this wave?
    y = 2 cos x/2



  • Is this wave sine or cosine?
    cosine
  • What is the amplitude?
    2
  • What is the period?
    1
  • What is the equation for this wave?
    y = 2 cos x



  • Is this wave sine or cosine?
    cosine
  • What is the amplitude?
    3
  • What is the period?
    2
  • What is the equation for this wave?
    y = 3 cos 2x



  • Is this wave sine or cosine?
    sine
  • What is the amplitude?
    3
  • What is the period?
    2
  • What is the equation for this wave?
    y = 3 sin 2x



  • Is this wave sine or cosine?
    sine
  • What is the amplitude?
    4
  • What is the period?
    2
  • What is the equation for this wave?
    y = 4 sin 2x


Notice that the next three graphs are SHIFTED UPWARDS!


  • What is the period of this wave?
    1
  • What is the amplitude of this wave?
    2
  • Is this wave sine or cosine?
    sine
  • What is the equation for this wave?
    y = 1 + 2 sin x



Notice that this wave completes 3 cycles in 2π.

  • What is the period of this wave?
    3
  • What is the amplitude?
    1
  • Is this wave sine or cosine?
    cosine
  • What is the equation for this wave?
    y = 2 + cos 3 x


Notice that this wave completes 1/2 cycle in 2π.

  • What is the period of this wave?
    1/2
  • What is the amplitude?
    2
  • Is this wave sine or cosine?
    sine
  • What is the equation for this wave?
    y = 2 + 2 sin x/2