How to Hand Graph a Cosine

Trigonometry, the mathematical study of triangles, would be lost without cosine. You use the cosine function to relate triangle side lengths and angles. Cosine is periodic just as a circle function is. When you graph the cosine function, you create a visual relationship between triangles and circles. The cosine graph tracks the changes in the x-coordinate of every point on the circumference of a circle. Each point on the circle circumference also serves as the vertex of a triangle embedded in the circle. As you draw the graph by hand, you will see the periodic nature of cosine unfold.

Things You'll Need

  • Ruler
  • Graph paper
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Instructions

    • 1

      Draw and label the horizontal and vertical axes of the graph. Use a ruler to draw the axes on the graph paper. The horizontal x-axis represents the argument of the cosine function and can be represented by x. The vertical y-axis represents the cosine function and can be represented by y, where y = cosine (x).

    • 2

      Mark and label the intervals on the x- and y-axes. Start the x-axis at 0 degrees and end at 360 degrees, using intervals of 10 degrees. Start the y-axis at -1 and end at +1. Use intervals of 0.2 for the y-axis.

    • 3

      Write a selection of the integers ranging from 0 to 360 on a piece of paper. Include 0, 180, and 360 in your selection. The more integers you use, the more accurate your graph will be.

    • 4

      Use the cosine function on your calculator to find the cosine of each integer and write it alongside the integer. Each pair of numbers (integer; cosine of integer) is a data point for your graph. The integer is the x-coordinate of the point, while the cosine of the integer is the y-coordinate of the point (x; y). For example, the cosine of the integer 100 is -0.17 = cosine (100). The resulting data point is (100; -0.17) = (x; y).

    • 5

      Plot all the (x, y) data points on the graph paper. Use a pencil dot to represent each point.

    • 6

      Draw a line through the data points. Avoid drawing the line from dot to dot. Rather, use a sweeping motion of the pencil across the paper so that the curved line passes smoothly through all the data points.

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