Introduction

We have encountered six trigonometric functions of an acute angle so far: , , , , , and . They all help up get information about right triangles having as one of the inner angles. But here is the thing: at this stage, they all carry the same information. All of these quantities are positive real numbers, and we have not only the Pythagorean Theorem, but also the fundamental relations

Recall that while the first one is the most important one, the second two are immediate consequences of the first, by dividing it by and , respectively. With all of this in place, once you have one of the six trigonometric values at , you can in fact find all of them. We’ll explore this in this section, with several examples.

How to find all trigonometric functions, given one of them?

There are a few main facts one should keep in mind here.

  • You know if and only if you know .
  • You know if and only if you know .
  • You know if and only if you know .
  • If you know and , you know .

And there are two strategies: using just the trigonometric identities and proceeding algebraically (let’s call this “strategy 1”), or drawing a suitable right triangle and thinking of , and (let’s call this “strategy 2”). We’ll illustrate both of them with several examples, but at the end of the day, you may choose whichever strategy you’d like (unless specifically instructed otherwise).

Let’s summarize the highlights of the strategy, from the algebraic perspective.

(a)
If you’re given or , use the fundamental identity to find the other one. Then find , and flip all the fractions to get , and .
(b)
If you’re given or , flip it to get or , and proceed as (a).
(c)
If you’re given , use to find . Once you have , you have . Then proceed as (a).
(d)
If you’re given , flip it to get , and proceed as (c).

Note that we’re employing a mathematician’s general philosophy here: take a problem and reduce it to something which you already know how to solve (namely, we’re arguing that — morally — if you know how to solve the problem when you were given either or , then you in fact know how to solve it when given any of the six trigonometric functions). And also from the geometric perspective, the strategy is even easier to describe: recognize the trigonometric function you were given in terms of , and , then draw a right triangle with this information. You will be missing one side, which can be found with the Pythagorean Theorem. Once you have all sides, you can find all the ratios between sides.

Of course, the two above ways to go about this are not the only ones, but they’re as good a recipe as any. In any case, you have room for creativity here. And even if one method seems easier than the other, it is useful to be comfortable with both, as this is already a good chance to start getting acquainted with trigonometry identities, which will be indispensable later.

We will see later how to define and deal with trigonometric functions for angles which are not necessarily acute. Then, everything we did here becomes slightly more subtle, as one must now pay attention to signs (for example, we’ll have that ). But the overall program of using the fundamental trigonometric identities and the relations between the main trigonometric functions (, , and ) with their reciprocals (, , and ) will always be useful.