Trigonometric Functions and Their Graphs:
The Sine and Cosine

At first, trig ratios related only to right triangles. Then you learned how to find ratios for any angle, using all four quadrants. Then you learned about the unit circle, in which the value of the hypotenuse was always r = 1 so that sin(θ) = y and cos(θ) = x. In other words, you progressed from geometrical figures to a situation in which there was just one input (one angle measure, instead of three sides and an angle) leading to one output (the value of the trig ratio). And this kind of relationship can be turned into a function.

Looking at the sine ratio in the four quadrants, we can take the input (the angle measure θ), "unwind" this from the unit circle, and put it on the horizontal axis of a standard graph in the x,y-plane. Then we can take the output (the value of sin(θ) = y) and use this value as the height of the function.

The Sine Wave

From the above graph, showing the sine function from –3π to +5π, you can probably guess why this graph is called the sine "wave": the circle's angles repeat themselves with every revolution, so the sine's values repeat themselves with every length of 2π, and the resulting curve is a wave, forever repeating the same up-and-down wave. (My horizontal axis is labelled with decimal approximations of π because that's all my grapher can handle. When you hand-draw graphs, use the exact values: π, 2π, π/2, etc.)

When you do your sine graphs, don't try to plot loads of points. Instead, note the "important" points. The sine wave is at zero (that is, on the x-axis) at x = 0, π, and 2π; it is at 1when x = π/2; it is at –1 when x = 3π/2. Plot these five points, and then fill in the curve.

We can do the same sort of function conversion with the cosine ratio:

The Cosine Wave

As you can see from the extended sine and cosine graphs, each curve repeats itself regularly. This trait is called "periodicity", because there is a "period" over which the curve repeats itself over and over. The length of the period for the sine and cosine curves is clearly 2π: "once around" a circle. Also, each of sine and cosine vary back and forth between –1 and +1. The curvesgo one unit above and below their midlines (here, the x-axis). This value of "1" is called the "amplitude".

When you graph, don't try to plot loads of points. Note that the cosine is at y = 1 when x = 0 and 2π; at y = 0 for x = π/2 and 3π/2, and at y = –1 for x = π. Plot these five "interesting" points, and then fill in the curve.

The Tangent

The next trig function is the tangent, but that's difficult to show on the unit circle. So let's take a closer look at the sine and cosines graphs, keeping in mind that tan(θ) = sin(θ)/cos(θ).

The tangent will be zero wherever its numerator (the sine) is zero. This happens at 0, π, 2π, 3π, etc, and at –π, –2π, –3π, etc. Let's just consider 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 be undefined wherever its denominator (the cosine) is zero. Thinking back to when you learned about graphing rational functions, a zero in the denominator means you'll have a vertical asymptote. So the tangent will have vertical asymptotes wherever the cosine is zero: at –π/2, π/2, and 3π/2. Let's put dots for the zeroes and dashed vertical lines for the asymptotes:

Now we can use what we know about sine, cosine, and asymptotes to fill in the rest of the tangent's graph: We know that the graph will never touch or cross the vertical asymptotes; we know that, between a zero and an asymptote, the graph will either be below the axis (and slide down the asymptote to negative infinity) or else be above the axis (and skinny up the asymptote to positive infinity). Between zero and π/2, sine and cosine are both positive. This means that the tangent, being their quotient, is positive, so the graph slides up the asymptote: Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved

Between π/2 and π, sine is positive but cosine is negative. These opposite signs mean that the tangent quotient will be negative, so it will come up the asymptote from below, to meet the x-axis at x = π:

Since sine and cosine are periodic, then tangent has to be, as well. A quick check of the signs tells us how to fill in the rest of the graph:

  • –π to –π/2: sine is negative and cosine is negative, so tangent is positive
  • –π/2 to 0: sine is negative but cosine is positive, so tangent is negative
  • π to 3π/2: sine is negative and cosine is negative, so tangent is positive
  • 3π/2 to 2π: sine is negative but cosine is positive, so tangent is negative

Now we can complete our graph:

The Tangent Graph

As you can see, the tangent has a period of π, with each period separated by a vertical asymptote. The concept of "amplitude" doesn't really apply.

For graphing, draw in the zeroes at x = 0, π, 2π, etc, and dash in the vertical asymptotes midway between each zero. Then draw in the curve. You can plot a few more points if you like, but you don't generally gain much from doing so.

If you prefer memorizing graphs, then memorize the above. But I always had trouble keeping straight anything much past sine and cosine, so I used the reasoning demonstrated above to figure out the tangent (and the other trig) graphs. As long as you know your sines and cosines very well, you'll be able to figure out everything else.

The Co-Functions

What about the co-functions, the secant, the cosecant, and the cotangent?

The cosecant is the reciprocal of the sine. Wherever the sine is zero, the cosecant will be undefined, so there will be a vertical asymptote. Wherever the sine reaches its maximum value of 1, the cosecant will reach its minimum value of 1; wherever the sine reaches its minimum value of –1, the cosecant will reach its maximum value of –1. Wherever the sine is positive but less than 1, the cosecant will be positive but greater than 1; wherever the sine is negative but greater than –1, the cosecant will be negative but less than –1.

So I'll lightly draw the sine wave...
Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved /
...I'll draw vertical asymptotes through its zeroes and note the min/max points... /
...and then I'll fill in the graph. /
The Cosecant Graph

By using the same reasoning with the cosine wave, I can create the secant graph:

The Secant Graph

The secant and cosecant have periods of length 2π, and we don't consider amplitude for these curves.

The cotangent is the reciprocal of the tangent. Wherever the tangent is zero, the cotangent will have a vertical asymptote; wherever the tangent has a vertical asymptote, the cotangent will have a zero. And the signs on each interval will be the same. So the cotangent graph looks like this:

The Cotangent Graph

The cotangent has a period of π, and we don't bother with the amplitude.

When you need to do the graphs, you may be tempted to try to compute a lot of plot points. But all you really need to know is where the graph is zero, where it's equal to 1, and / or where it has a vertical asymptote. If you know the behavior of the function at zero, π/2, π, 3π/2, and 2π, then you can fill in the rest. That's really all you "need".