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Graphs of Sine and Cosine Functions
Introduction: In this lessons, the basic graphs of sine and cosine will be discussed and illustrated.

The Lesson:
The graphs of have some basic features which makes them easy to identify and accurately diagram. The domain, the possible values for x, is all real numbers. We will use radian measure so that any real number can be used for x. This is because any real number can be associated with an arc length on a circle of radius 1, the definition of a radian measure. Therefore, when using a calculator, set the mode for radians, not degrees.

We first construct a table of some common values for sin(x) and plot those points.

x0
sin(x)00.50.7070.86610.8660.707


These points are plotted below. The WINDOW is X: and Y: (– 2, 2, 1).

Since the domain of is all real numbers, we can continue plotting points both to the left and right of the ones shown. Connecting those points gives the graph shown below, which contains the original points plotted.

All sine and cosine graphs will have this general shape which is called a sinusoid. If we focus on the important points which are the maximum, minimum and x-intercepts we can sketch a sinusoid containing those points quite accurately. The graph above of shows a maximum value of 1 at , a minimum of – 1 at and x-intercepts at .

The graph of , shown below, is also a sinusoid. The WINDOW is X: and Y: (– 2, 2, 1). This graph shows a maximum value of 1 at 0 and , a minimum of – 1 at and x-intercepts at .

Facts:
  1. The sinusoid graph of or is a continuous repetition of one wave. For cosine, one complete wave can be seen from maximum point (0, 1) to the next maximum point at . For sine, one complete wave can be seen from an x-intercept at (0, 0) to the x-intercept at .
  2. The length (distance along the x-axis) of one complete wave is called the period of the wave. In the case of of and both and the period is . Functions with periods other than are discussed in the lesson on period and frequency.
  3. The height or amplitude of the wave is the vertical distance from the middle of the wave to a maximum point or minimum point. For both and the amplitude is 1.
Let's Practice:
  1. What is the graph of ?
The graph is shown below using a WINDOW X: and Y: (– 2, 2, 1). The only difference between this graph and is that the 2 has made the graph go twice as high and twice as low. Therefore the amplitude is 2.

  1. What is the graph of ?
The graph is shown below using a WINDOW X: and Y: (– 3, 3, 1). One difference between this graph and is that the 3 has made the graph go three times as high and three times as low. The amplitude is 3. The other difference is caused by the negative sign. This causes the graph to be reflected or “flipped” over the x-axis. Where had a maximum point at (0, 1), when x = 0, has a minimum point at (0, – 3).


Examples
Example Describe the graph of ?
What is your answer?
 
Example Describe the graph of ?
What is your answer?
 



M Ransom

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