Precalculus Notes, Trig 3

There are two very special right triangles which were studied in geometry that give us special relationships. The first of these is the isosceles right triangle. Because the two legs are of equal length, it can be easily shown that the hypotenuse of this right triangle is times either leg. Because we know this, we know the ratios of all the basic trig functions. See below.

The second very special right triangle we will consider is the 30-60-90 right triangle. This triangle is special because of the geometry theorem: The leg opposite the 30 degree angle in a right triangle is one half of the hypotenuse. Again, the Pythagorean Theorem allows us to come up with simple ratios for each of the basic trig functions. Not only does this give us the 3 trig functions for 30 degrees, it also gives us the 3 trig functions for 60 degrees. (the other acute angle of the right triangle.) See below. These two special triangles and their resulting ratios are extremely important and need to be memorized.

It is also important that we begin to see patterns emerging for what is happening to each trig function as angles gradually increase from 0 to 360 degrees. The chart below I call a behavior chart for the sine function. Because we know that the sine is defined as the leg opposite the angle divided by the hypotenuse we should be able to see that the SIN function is the vertical leg of the multiple right triangles formed in the unit circle below. Please not that the hypotenuse of the triangles below has a length of 1. So, by watching the length and position of the vertical leg of the multiple right triangles below, we can observe the general behavior of the sine function.

At zero degrees, the triangle is completely collapsed vertically, and the vertical length is of height zero. Therefore, the sine of zero degrees is zero. As the acute angles gradually increase you should be able to see two more things: 1) the sine is increasing in the first quadrant, and 2) at 90 degrees (Pi/2 radians) the triangle is completely compressed horizontally and the vertical leg length will be 1. So the sine of 90 degrees is 1.

In the second quadrant you can see that the vertical leg lengths will be decreasing, and will eventually reduce back to zero when the angle is 180 degrees (Pi radians).

In the third and fourth quadrants, because the vertical leg length is now below the x-axis, we think of its length as negative. So in quadrant 3 the sine goes from zero to -1 and is therefore decreasing. In the fourth quadrant, it goes from -1 to 0, and is therefore increasing.

A mental image of this rotating angle and what happens to this vertical leg is vital to knowing what happens to the sine function.

Evaluation at Special Angles @ WLC

So now we can begin to develop results mentally of the three basic trig functions at the quandrantal angles (0, 90, 270, 360) and the special angles of 30, 45, and 60. This chart will grow as we continue our study.

The Trig You Need for Calculus @ WLC