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Trigonometric Identities - Problem 2

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 opposite angle identities. First let’s review these definitions a function is even if f of -x equals f of x,a function is add is f of -x equals -1/2 of x. What does that mean?

This says that if you take opposite inputs you get the same output. That’s what it means to be an even function and an odd function is a function where opposite inputs give opposite outputs. So we’ll see that actually all the trig functions are either even or odd that’s what we are about to show.

So let’s take a look at the unit circle. Here’s the unit circle with angle theta drawn on it and this is point p(x,y) and you know that x gives me the cosine of theta and y gives me the sine of theta. So I want to find out what the cosine of negative theta is. And I’ve drawn negative theta.

Negative theta will be a reflection of positive theta across the x axis and so the coordinates of this point q which is a reflection of point p, would be the same for x and opposite for y. And that means the cosine of negative theta is exactly the same as the cosine of theta. So we just proved that cosine is even.

Now what about sine? Well the y values are opposite so the sine of negative theta is minus y and y is the sine of theta. Sine is not function.

Now what about tangent? Tangent, remember the identity tangent theta is sine theta over cosine theta. Tangent of negative theta would be sine of negative theta over cosine negative theta. Now sine of negative theta is the opposite of sine theta and cosine of negative theta is cosine theta, let me write that down. Negative sine theta cosine theta and there we have it. That’s exactly minus tangent theta.

And so tangent is not a function. Like I said all the trig functions are either even or odd cosine is even, sine is odd and tangent is odd this would be really important identities later on.

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