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Trigonometric Functionsand their Graphs:The Tangent(page 2 the 3)

Sections: The sine and cosine, The tangent, The co-functions




You are watching: What is the period of a tangent function

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The following trig role is the tangent, but that"s difficult to display on the unit circle. Therefore let"s take a closer look in ~ the sine and cosines graphs, keeping in mind the tan(θ) = sin(θ)/cos(θ).

The tangent will certainly be zero wherever its molecule (the sine) is zero. This happens at 0, π, 2π, 3π, etc, and at –π, –2π, –3π, etc. Let"s just take into consideration the region from –π to 2π, for now. So the tangent will be zero (that is, it will cross the x-axis) at –π, 0, π, and 2π.

The tangent will certainly be undefined wherever that denominator (the cosine) is zero. Thinking ago to as soon as you learned around graphing reasonable functions, a zero in the denominator way you"ll have actually a upright asymptote. So the tangent will have vertical asymptotes where the cosine is zero: in ~ –π/2, π/2, and also 3π/2. Let"s placed dots for the zeroes and also dashed vertical lines because that the asymptotes:

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Between π/2 and also π, sine is positive yet cosine is negative. This opposite signs mean the the tangent quotient will certainly be negative, so it will certainly come up the asymptote indigenous below, to accomplish the x-axis in ~ x = π:

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Since sine and cosine room periodic, climate tangent has to be, as well. A quick check of the signs tells us exactly how to to fill in the remainder of the graph:

–π come –π/2: sine is an unfavorable and cosine is negative, for this reason tangent is positive –π/2 come 0: sine is an unfavorable but cosine is positive, for this reason tangent is an unfavorable π to 3π/2: sine is an unfavorable and cosine is negative, therefore tangent is positive 3π/2 to 2π: sine is an adverse but cosine is positive, so tangent is an unfavorable

Now us can finish our graph:

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The Tangent Graph

As you have the right to see, the tangent has a period of π, with each duration separated by a upright asymptote. The ide of "amplitude" doesn"t really apply.

For graphing, draw in the zeroes in ~ x = 0, π, 2π, etc, and also dash in the vertical asymptotes midway in between each zero. Then attract in the curve. You deserve to plot a couple of more points if girlfriend like, but you don"t generally obtain much indigenous doing so.

If you prefer memorizing graphs, then memorize the above. However I always had trouble keeping straight anything much past sine and cosine, so I offered the reasoning demonstrated over to number out the tangent (and the other trig) graphs. As lengthy as you understand your sines and also cosines an extremely well, you"ll be able to figure out everything else.

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Cite this write-up as:

Stapel, Elizabeth. "Trigonometric Functions and Their Graphs: Tangent." ptcouncil.net. Easily accessible from https://www.ptcouncil.net/modules/triggrph2.htm. Accessed 2016