Graphing Sine, Cosine, and Tangent
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Introduction

    The graphs of trigonometric functions have several properties to elicit. To be able to graph these functions by hand, we have to understand them. Before we progress, take a look at this video that describes some of the basics of sine and cosine curves.

      ideo: Graphing Basics: Sine and Cosine


    The video in the previous section described several parameters. This section will define them with precision within the following table.



Amplitude and Period

    To calculate amplitude and period, the equation of our sine and cosine curves have to be in a specific form. The equations have to look like this.

    Once in that form, the parameters for amplitude and period are calculated as follows.



Amplitude and Period

    Here is a sine function we will graph.

    The a-value is the number in front of the sine function, which is 2. This makes the amplitude equal to |2| or simply 2. The graph of the function has a maximum y-value of 2 and a minimum y-value of -2.

    The b-value is the number next to the x-term, which is 3. This means the period is 360 degrees divided by 3 or 120. So, the curve has a y-intercept of zero (because it is a sine curve it passes through the origin) and it completes one cycle in 120 degrees.

    This is the graph of the sine curve.

    This particular interval of the curve is obtained by looking at the starting point (0,0) and the end point (120,0). The domain (the x-values) of this cycle go from 0 to 120. So, we write this interval as [0,120].

    This video will demonstrate how to graph a different sine function with two parameters: amplitude and period.

      ideo: Graphing Sine: Amplitude and Period
      uiz: Graphing Sine & Cosine: Amplitude and Period

Amplitude and Period

   Here is a cosine function we will graph.

    The a-value is the number in front of the sine function, which is 4. This makes the amplitude equal to |4| or 4. The graph of the function has a maximum y-value of 4 and a minimum y-value of -4.

    The b-value is the number next to the x-term, which is 2. This means the period is 360 degrees divided by 2 or 180. So, the curve has a y-intercept at its maximum (0,4) (because it is a cosine curve) and it completes one cycle in 180 degrees.

    This is the graph of the cosine curve.

    This particular interval of the curve is obtained by looking at the starting point (0,4) and the end point (180,4). The domain (the x-values) of this cycle go from 0 to 180. So, we write this interval as [0,180].

    This video will demonstrate how to graph a different cosine function with two parameters: amplitude and period.

      ideo: Graphing Cosine: Amplitude and Period
      uiz: Graphing Sine & Cosine: Amplitude and Period

Phase Shift and Vertical Shift

    To calculate phase shift and vertical shift, the equation of our sine and cosine curves have to be in a specific form. The equations have to look like this.

    Once in that form, all the parameters can be calculated as follows.

    Notice that the equations have subtraction signs inside the parentheses. The c-values have subtraction signs in front of them. However, the phase shift is the opposite. This will be demonstrated in the next two sections.

Amp, Period, Phase Shift, and Vert. Shift

    This video will demonstrate how to graph a sine function with four parameters: amplitude, period, phase shift, and vertical shift.

      ideo: Graphing Sine: Amplitude, Period, Phase Shift, and Vertical Shift
      uiz: Graphing Sine/Cosine: Amplitude, Period, Phase Shift, and Vertical Shift

Amp, Period, Phase Shift, and Vert. Shift

    This video will demonstrate how to graph a cosine function with four parameters: amplitude, period, phase shift, and vertical shift.

      ideo: Graphing Cosine: Amplitude, Period, Phase Shift, and Vertical Shift
      uiz: Graphing Sine/Cosine: Amplitude, Period, Phase Shift, and Vertical Shift

Period and Phase Shift

    This video will demonstrate how to graph a tangent function with two parameters: period and phase shift.

      ideo: Graphing Tangent: Period and Phase Shift


    Here are activities replated to the lessons in this section.

      ctivity: Graphing Trig Functions [amplitude, period]
      ctivity: Graphing Trig Functions 2 [amplitude, period, phase shift, horizontal shift]
      ctivity: Graphs to Equations [amplitude, period, interval, y-intercept, equation]


    Try this advanced lesson, which is closely related to the lessons above.

      esson: Graphing Lines
      esson: Graphing Linear Inequalities
      esson: Graphing Conic Sections
      esson: Linear Programming