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There are six major trigonometric functions, at least three of which you have probably heard: sine, cosine, and tangent. All of the trigonometric functions (even those that are offended because you haven’t heard of them) are periodic functions. In this guide we’ll explain what a periodic function is, and look at how each of the trigonometric functions work.

A periodic function has the unique characteristic that it repeats itself after some fixed period of time. Think of the rising of the sun as a periodic function—every 24 hours (a fixed amount of time) the sun appears on the horizon.

The amount of horizontal space it takes until the function repeats itself is called the period. For the most basic trigonometric functions (sine and cosine), the period is 2*π.* Look at this graph of one period of *y* = sin *x.*

The graph of the sine function is a wave, reaching a maximum height of 1 and a minimum height of 1. On the piece of the graph shown above, the maximum height is reached at and . The distance between these two points, where the graph repeats its value, is 2*π*. If that doesn’t help you understand what is meant by period, consider the darkened portion of the graph.

This piece begins at the origin (0,0) and wiggles up and down, returning to a height of 0 when *x* = 2*π*. True, the graph hits a height of 0, repeating its value, when *x* = *π*, but it hasn’t completed its period yet—that is only finished at *x* = 2*π*.

If you were to extend the graph of the sine function infinitely right and left, it would redraw itself every 2*π*. Because of this property of periodic functions, you can list an infinite number of inputs that have identical sine values. These are called coterminal angles, and the next example focuses on them.

**Example 1:** List two additional angles (one positive and one negative) that have the same sine value as .

**Solution:** We know that sine repeats itself every 2*π*, so exactly 2*π* further up and down the *x*-axis from , the value will be the same. To find these values, simply add 2*π* to in order to get one and subtract 2*π* from to get the other. In order to add and subtract the values, you’ll have to get common denominators:

Therefore, the angles and are coterminal to and .

The sine function is defined for all real numbers, and this unrestricted domain makes the function very trustworthy and versatile. The range is 1 ≤ *y* ≤ 1, so all sine values fall within those boundaries. Notice that the sine function has a value of 0 whenever the input is a multiple of *π*. Sometimes, people get confused when memorizing unit circle values (more on the unit circle later in this chapter). If you remember the graph of sine, you can easily remember that sin 0 = sin *π* = sin 2*π* = 0, because that’s where the graph crosses the *x*-axis. The period of the sine function is 2*π*.

Cosine is the “cofunction” of sine. (In other words, their names are the same, except one has a “co-” prefix, but I bet you figured that out.) As such, it looks very similar, possessing the same domain, range, and period. In fact, if you shift the entire graph of *y* = cos *x* a total of radians to the right, you get the graph of *y* = sin *x*! The cosine has a value of 0 at all the “half-*π*’s,” such as and .

The tangent is defined as the quotient of the previous two functions: . Thus, to evaluate tan , you’d actually evaluate (which will equal 1 for those of you who are curious, but more about that later). Because the cosine appears in the denominator, the tangent will be undefined whenever the cosine equals 0, which (according to the last section) is at the half-*π*’s. Notice that the graph of the tangent has vertical asymptotes at these values. The tangent equals 0 at each midpoint between the asymptotes. The domain of the tangent excludes the “half-*π*’s,” , but the range is all real numbers. The period of the tangent is *π*—notice that there’s a full copy of one tangent period between and .

The cofunction of tangent, cotangent, is the spitting image of tangent, with a few exceptions. It, too, is defined by a quotient: . In fact, the cotangent is technically the reciprocal of the tangent, so you can also write . Therefore, this function is undefined whenever sin *x* = 0, which occurs at all the multiples of *π*: {…, 2*π*,*π*,0,*π*,2*π*,…}, so the domain includes all real numbers except that set. The range, like that of the tangent, is all real numbers, and the period, *π*, also matches tangent’s.

The secant function is simply the reciprocal of cosine, so . Therefore, the graph of the secant is undefined (has vertical asymptotes) at the same places (and for the same reasons) as the tangent, since they both have the same denominator. Hence, the two functions also have the same domain. Notice that the secant has no *x*-intercepts. In fact, it doesn’t even come close to the *x*-axis, only venturing as far in as 1 and 1. That’s a fascinating comparison: the cosine has a range of 1 ≤ *y* ≤ 1, but the secant has a range of *y* ≤ 1 or *y* ≥ 1—almost the exact opposite. Because secant is based directly on cosine, the functions have the same period, 2*π*.

Very similar to its cofunction sister, this function has the same range and period as the secant, differing only in its domain. Because the cosecant is defined as the reciprocal of the sine, , cosecant will have the same domain as cotangent, as they share the same denominator.

In essence, four of the trig functions are based on the other two (sine and cosine), so those two alone are sufficient to generate values for the rest.

**Example 2:** If and , evaluate tan *q* and sec *q*.

**Solution:** Let’s tackle these one at a time. First of all, you know that , so:

Multiply the top and bottom by 3 to simplify the fraction:

Now, on to sec *q*—this is even easier. Because you know that , and (because the secant is the reciprocal of the cosine):

Now that you know how trigonometric functions work, hopefully the rest of your trig studies will make more sense. Good luck!

From *The Complete Idiot’s Guide to Calculus, Second Edition,* by W. Michael Kelley