Graphs of $$\sec(x)$$, $$\cosec(x)$$ and $$\cot(x)$$

In a nutshell

By using the graphs of $$y=\sin(x)$$, $$y=\cos(x)$$ and $$y=\tan(x)$$, you can sketch a useful plot of the reciprocal graphs $$y=\cosec(x)$$​, $$y=\sec(x)$$ and $$y=\cot(x)$$.

The functions

Recall that 

​$$\begin{aligned}\cosec(x)&=\frac1{\sin(x)}\\\sec(x)&=\frac1{\cos(x)}\\\cot(x)&=\frac1{\tan(x)}\end{aligned}$$​​

This helps when using the plots of the graphs of $$y=\sin(x)$$, $$y=\cos(x)$$ and $$y=\tan(x)$$.

The graphs

When a graph has an $$x$$-intercept, it follows that the reciprocal of that graph has an asymptote, and vice versa. Below, the graphs are given using degrees. The shapes would be the same for radians, but the $$x$$-axes would be in radians.​

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​​$$y=\cosec(x)$$

The domain of this function is all the real numbers, excluding integer multiples of $$180$$ (in degrees) or multiples of $$\pi$$ (in radians). At these points, there are asymptotes. The range is $$y\leq-1$$ or $$1\leq y$$. As with $$y=\sin(x)$$, it has a period of $$360$$ (if in degrees) or $$2\pi$$ (if in radians). Notice that if you drew $$y=\sin(x)$$ on the same grid, the peaks of $$y=\cosec(x)$$ would meet the troughs of $$y=\sin(x)$$, and vice versa.​

$$y=\sec(x)$$​​

The domain of this function is all the real numbers, excluding odd integer multiples of $$90$$ (in degrees) or odd multiples of $$\frac{\pi}2$$ (in radians). At these points, there are asymptotes. The range is $$y\leq-1$$ or $$1\leq y$$. As with $$y=\cos(x)$$, it has a period of $$360$$ (if in degrees) or $$2\pi$$ (if in radians). Notice that if you drew $$y=\cos(x)$$ on the same grid, the peaks of $$y=\sec(x)$$ would meet the troughs of $$y=\cos(x)$$, and vice versa.​

​$$y=\cot(x)$$​​

The domain of this function is all all real numbers, excluding integer multiples of $$180$$ (in degrees) or integer multiples of $$\pi$$ (in radians): this is where there are asymptotes. The range is all real numbers.