In trigonometry, multiplying the angle variable in a tangent function has the same effect as it does with sine and cosine functions — it affects the period of the function. If the multiple is 2, as in *y* = tan 2*x*, then the tangent function makes twice as many cycles in the usual amount of space. In other words, the period is *p*/2, which is the tangent’s usual period, *p*, divided by 2.

The preceding figure shows a few graphs to illustrate the effect of multiplying the angle variable by a number greater than 1 and then by a number between 0 and 1.

The graph of *y* = tan 3*x* doesn’t show all the asymptotes, but that graph has three times as many tangent curves as usual. The graph of

only half as many cycles — or it takes twice as long to complete one cycle.