A Different View

Consider the following problem. Let be an angle between 90° and 180°. Assume . Find the values of and .

At first glance, you might say, ``But I cannot find the values of the other trig functions because I do not know what the angle equals.'' You do not need to know the actual value of in order to find and . All you need to do is to draw the diagram corresponding to the information given in the statement of the problem.

First, because you know that is an angle between 90° and 180°, you know that the terminal side of lies in Quadrant II. Second, since , you see that you can write in the values of both the hypotenuse of the right triangle in question as well as the opposite leg since is defined as

Thus, your drawing should look something like this:

Now, all you need is the length of the adjacent leg. Remember that, since your point has a negative value for x, we will label this with a negative sign. How do you find the length of this third leg? Answer: The Pythagorean Theorem! If we call the length of the adjacent leg x, then we have x² + 1² = 5² from the Pythagorean Theorem. Hence, we have x² + 1 = 25 or x² = 24. Therefore, we know . In this case, we choose or since, as we noted above, we need to label this leg with a negative value. Therefore, our diagram now can be completed as follows:

From this we see that