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Trigonometric Identities - Problem 3

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 the periodicity of sine cosine and tangent. I got the unit circle drawn here remember that x equals cosine theta and y equals sine theta. I’ve drawn two angles angle theta and angle theta plus 2 pi remember one revolution on the circle is 2 pi.

Theta and theta plus 2pi are we call co-terminal angles they both have the same terminal sides. And plus they both end at the same point p of (x,y) so cosine of theta plus 2pi is going to equal x which is the cosine of theta. Now that means cosine is periodic with period 2pi and sine has the same property. And you may already know this but this is a way of showing it.

So again theta theta plus 2 pi we have the same y coordinate so both the sine of theta plus 2pi and sine theta are equals to y. What about tangent, let’s take a look at this unit circle I’ve drawn the angle theta and I’ve drawn the angle theta plus pi. Notice that I’ve gone an extra 180 degrees compare the points p and q, point p in this picture is in the first quadrant both coordinates will be positive point q is in the third quadrant where both coordinates will be negative. So both coordinates are opposites for this point.

So what would the tangent of theta plus pi p? It would be sine of theta plus pi over cosine of theta plus pi. Now sine of theta plus pi is -y. And cosine of theta plus pi is -x and that’s exactly the same as y over x. And that’s by definition the tangent of theta.

So if tangent of theta plus pi equals tangent theta that means that tangent has a period of pi so tangent is going to repeat itself every pi sine and cosine are going to repeat themselves every 2pi. Those are the periodicity identities.

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