Graphs of Elementary Trigonometric Functions

To see the functions in a clear manner, we’ll graph the trigonometric functions on the same graph as their reciprocals

This means that the $sine$ curve and the $cosecant$ curve are graphed on the same graph.

It also means that the $cosine$ curve and the $secant$ curve are graphed on the same graph.

And finally, it means that the $tangent$ curve and the $cotangent$ curve are graphed on the same graph.

Graph of:

$f(x)=\sin x$

$g(x)=\csc x$

We notice that the two functions meet at the maximum and minimum values.

The function $g(x)=\csc x$ is not defined when $\sin x=0$ . These are called the vertical asymptotes for the function $g(x)=\csc x$

Graph of:

$f(x)=\cos x$

$g(x)=\sec x$

We also notice that the two functions meet at the maximum and minimum values.

The function $g(x)=\sec x$ is not defined when $\cos x=0$ . These are again the vertical asymptotes for the function $g(x)=\sec x$

Graph of:

$f(x)=\tan x$

$g(x)=\cot x$

We also notice that the two functions intersect only when $\sin x=\cos x$ .

These points are multiple of a single angle when $x=\frac{\pi}{4}+n\pi$ . Each function has an asymptote when the other function is $0$ .

We can see in these graphs that each function resumes its pattern after some values of $x$ . They are said to be periodic.

The repeated pattern is called a $cycle$ .

Let’s get a summary of the general behavior of these functions

The “sine” function:

$f(x)=A\sin (Bx+C)+D$

With A,B,C and D being constants, we can see that we can easily graph this complex function using the base function of $f(x)=\sin x$ and the technics we learned from the quadratic functions.

The Amplitude:

The amplitude $A$ is the measure of the maximum or minimum values from the midline $D$ .

The amplitude itself is always positive. $|A|$ . The original $\sin x$ is multiplied by $A$ .

Period:

This is the horizontal width of a single cycle or wave. After this wave, the pattern repeats itself.

We can see that functions $y=\sin x$ , $y=\csc x$ , $y=\cos x$ and $y=\sec x$ repeat after $2\pi$ .

For the function:

$y=A\sin (Bx+C)+D$ , we know that the period of the original/parent function $y= \sin x$ is $2\pi$

The period of this function will be:

$period=\frac{2\pi}{|B|}$

Frequency:

The frequency of the function is simply the reciprocal of the period.

$frequency=\frac{1}{period}$

The phase shift:

This is the horizontal shift of the original function and is calculated using:

$phase\; shift=-\frac{C}{B}$

The vertical shift:

The vertical shift is simply $D$

All these values will be calculated the same way when we graph the function:

$f(x)=A\cos (Bx+C)+D$

The period is useful when solving trigonometric equations.

For tangent and cotangent functions, we know that the period of the original/ parent function is $\pi$ .

The phase shift is important for the tangent:

$phase\; shift=-\frac{C}{B}$ . The first cycle begins at the ‘zero’ that is “Phase shift units to the right of the origin”. The asymptotes are at $\frac{1}{2}P$ to the left of the beginning cycle, and another at $\frac{1}{2}P$ to the right of the beginning cycle.