Experience First

Students will use what they know about tangent to plot values on a coordinate plane.  You may have to help them with the shape.  They should use their knowledge of the undefined locations to draw vertical asymptotes and then plot the points between to sketch the curve.  Have students who have the general shape draw their graphs on the board for the other students to see.

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Formalize Later

The main takeaway is that the tangent function has asymptotes where the cosine function equals 0 and the cotangent function has asymptotes where the sine function equals 0.  It’s also important for the students to think about the period and range of tangent and cotangent.  These functions have half the period because the outputs repeat in half the length of the sine and cosine functions.  We typically ask our students to draw at least 2 periods of the tangent and cotangent functions so they’re still graphing on the same intervals as sine and cosine.

The range is all real numbers because the outputs get very close to the asymptotes and approach negative and positive infinity on either side.  Remind them of the ratios used to develop tangent and cotangent and how there are no restrictions for the ratios between the opposite and adjacent sides of a right triangle.

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