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- 1). Determine the zeros for the equivalent function sin(x) / cos(x). For example, since the basic tangent function equals sin(x) / cos(x) which equals zero when sin(x) = 0, and considering only the region 0 to 2PI, it is seen that sin(x) = 0 at 0, PI and 2PI.
- 2). Plot the zeroes on a coordinate plane. These points are where the tangent function passes through the x-axis.
- 3). Determine where the tangent function is undefined. For example, the basic tangent function equivalent to sin(x) / cos(x) is undefined when cos(x) is equal to zero. Restricted to an interval of 0 to 2PI, cos(x) = 0 at (PI / 2) and (3PI / 2).
- 4). Plot the zeroes of cos(x) as asymptotes of the tangent function. These are denoted as dotted vertical lines where the function is undefined.
- 5). Graph the tangent by recognizing that it will never cross or touch the asymptotes and that the sign of the tangent function will be the same to the sign of sin(x) / cos(x). Between 0 and (PI / 2) both sine and cosine are positive, so the graph is also positive. The graph moves up from 0 and rises toward infinity, becoming increasingly closer to the asymptote. The graph has the shape of the right side of a "U."
- 6). Continue to graph the function recognizing that between (PI / 2) and PI the tangent function is negative, and between PI and (3PI / 2) the tangent is positive. The graph between (PI / 2) and (3PI / 2) rises from negative infinity at the asymptote, through the x-axis, back up towards positive infinity at the asymptote at (3PI / 2).
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