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The Definitions of Sine and Cosine - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

I want to talk about some special values of sine and cosine; they’re called the unit circle definitions of sine and cosine. If you have a unit circle drawn and an angle drawn in standard position then the coordinates of this point P, x and y and the cosine and sine of angle theta. So x equals cosine theta and y equals sine theta.

In this problem I want to evaluate sine and cosine at what I call the quadrantal angles. Zero pi over 2, pi, 3 pi over 2 and 2 pi and there are lots of others but I particularly want to do the quadrantal angles between zero and 2 pi because they are really useful in graphing sine and cosine later on.

First of all, the angle I’ve got shown here looks like pi over 2 and remember pi is 180 degrees so pi over 2 is 90 degrees. But I want to start with theta equals zero and when theta equals zero, point P is going to be right here a the point (1,0) an so if x is 1 ad y is zero, then cosine of zero is 1 and sine of zero is zero. Let me fill that in on my table.

Now let’s do the situation that’s depicted, if theta is pi over 2, then the coordinates at this point are going to be (0,1). X will be zero and y will be 1. So the cosine of pi over 2 will be zero and the sine of pi over 2 will be 1.

At pi, point P is down here. So the x coordinate is going to be negative 1, y will be zero. And its cosine is negative 1, sine is zero. 3 times pi over 2, that 3 times 90 degrees so we’re down here at this point. The coordinates are zero, negative 1, so cosine is zero, sine is negative 1.

Finally all the way around, 2 pi radians bring me back to here, coordinates are again (1,0) so cosine of 2 pi is 1, sine of 2 pi is zero. We’re right back where we started. You can kind of see that as you keep advancing, sine and cosine are going to repeat their values over and over again. Their periodic functions and they repeat themselves every 2 pi. We’ll talk more about this later.

But anyway these values I have in my table are very, very important for graphing the sine and cosine functions as we’ll see later and so you want to make sure that you’re familiar with them or at least if you draw this picture, make sure that you can arrive at the values for sine and cosine of the quadrantal angles very quickly.

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