Analyzing the Graphs of y = sec x and y = cscx and Their Variations

The secant was defined by the reciprocal identity [latex]\sec x=\frac{1}{\cos x}[/latex]. Notice that the function is undefined when the cosine is 0, leading to vertical asymptotes at [latex]\frac{\pi}{2},\frac{3\pi}{2}\text{, etc}[/latex]. Because the cosine is never more than 1 in absolute value, the secant, being the reciprocal, will never be less than 1 in absolute value.

We can graph [latex]y=\sec x[/latex] by observing the graph of the cosine function because these two functions are reciprocals of one another. The graph of the cosine is shown as a dashed orange wave so we can see the relationship. Where the graph of the cosine function decreases, the graph of the secant function increases. Where the graph of the cosine function increases, the graph of the secant function decreases. When the cosine function is zero, the secant is undefined.

The secant graph has vertical asymptotes at each value of x where the cosine graph crosses the x-axis; we show these in the graph below with dashed vertical lines, but will not show all the asymptotes explicitly on all later graphs involving the secant and cosecant.

Note that, because cosine is an even function, secant is also an even function. That is, [latex]\sec(−x)=\sec x[/latex].

Graph of the secant function, [latex]f(x)=\sec x=\frac{1}{\cos x}[/latex]

As we did for the tangent function, we will again refer to the constant |A| as the stretching factor, not the amplitude.

Similar to the secant, the cosecant is defined by the reciprocal identity [latex]\csc x=1\sin x[/latex]. Notice that the function is undefined when the sine is 0, leading to a vertical asymptote in the graph at 0, π, etc. Since the sine is never more than 1 in absolute value, the cosecant, being the reciprocal, will never be less than 1 in absolute value.

We can graph [latex]y=\csc x[/latex] by observing the graph of the sine function because these two functions are reciprocals of one another. The graph of sine is shown as a dashed orange wave so we can see the relationship. Where the graph of the sine function decreases, the graph of the cosecant function increases. Where the graph of the sine function increases, the graph of the cosecant function decreases.

The cosecant graph has vertical asymptotes at each value of x where the sine graph crosses the x-axis; we show these in the graph below with dashed vertical lines.

Note that, since sine is an odd function, the cosecant function is also an odd function. That is, [latex]\csc(−x)=−\csc x[/latex].

The graph of cosecant, is similar to the graph of secant.

The graph of the cosecant function, [latex]f(x)=\csc x=\frac{1}{\sin x}/latex]

features of the graph of [latex]y=A\csc(Bx)[/latex]

• The amplitude is |A|.
• The period is [latex]\frac{2\pi}{|B|}[/latex].
• The domain is [latex]x\ne\frac{\pi}{|B|}k[/latex], where k is an integer.
• The range is [latex]( −\infty, −|A|] \cup [|A|, \infty)[/latex].
• The asymptotes occur at [latex]x=\frac{\pi}{|B|}k[/latex], where k is an integer.
• [latex]y=A\csc(Bx)[/latex] is an odd function because sine is an odd function