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.

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The three cofunction identities are useful because they can be used to convert, for example, sine into cosine or any trig function into its cofunction. When **graphing reciprocal trigonometric functions**, first find the values of the original trig function. Take the reciprocal of each value and plot the ordered pair in the coordinate plane.

I'm getting ready to graph the reciprocal trig functions but before I do it I need to prove a couple of identities.

First of all the co-function identities. Now the co-function identities are identities for sine of pi over 2 minus theta, cosine of pi over 2 minus theta and tangent of pi over 2 minus theta. So I've drawn the unit circle for pi over 2 minus theta and for theta. Now let's say that theta's an angle that terminates on this on this ray and the ray passes through point p with coordinates xy. Now point q is a reflection of point p in the line y=x and so its coordinates are going to be the reverse of the coordinates of point q point p rather. That means that the cosine of pi over 2 minus theta is going to be y and y is the sine of theta.

Similarly the sine of pi over 2 minus theta is x and x is the cosine of theta. Now once you have the 2 co-function identities for sine and cosine, you can find the one for tangent. Tangent pi over 2 minus theta is sine pi over 2 minus theta over cosine pi over 2 minus theta. Sine of pi over 2 minus theta is cosine and cosine of pi over 2 minus theta is sine. And that makes the tangent of pi over 2 minus theta cotangent theta. So what this means is one way to convert from sine to cosine is take the sine of pi over 2 minus theta.

Similarly the sine of pi over 2 minus theta is x and x is the cosine of theta. Now once you have the 2 co-function identities for sine and cosine, you can find the one for tangent. Tangent pi over 2 minus theta is sine pi over 2 minus theta over cosine pi over 2 minus theta. Sine of pi over 2 minus theta is cosine and cosine of pi over 2 minus theta is sine. And that makes the tangent of pi over 2 minus theta cotangent theta. So what this means is one way to convert from sine to cosine is take the sine of pi over 2 minus theta. You can convert from any trig function to its co-function by doing this. So this is an identity that's true for all 6 trig functions but I I just need them for sine, cosine and tangent.

Now the other identities I need to to prove are identities for sine of theta plus pi, cosine of theta plus pi and tangent of theta plus pi. And I've drawn theta and theta plus pi on the inner circle here. Now the coordinates of point q which is on the termina- terminating side of theta plus pi or -x -y, where x and y are the coordinates of point p. So the sine of theta plus pi is going to be -y and -y is the opposite of the sine of theta. The cosine of theta plus pi is going to be -x and -x is the opposite of the cosine of theta. And the tangent od theta plus pi is going to be this y coordinate divided by this x coordinate, -y over -x which is y over x which is tangent theta.

So the interesting thing here is that for sine and cosine, when you add pi you get the opposite of the sine value. Let's use this in an example. Suppose I want to complete a table of values and all I know for the values of cosine are these 4. I can continue this as far forward as I want by adding pi. Let me show you what I mean. If I add pi to minus pi over 2 over 3, I get 2 pi over 3. And the cosine of 2 pi over 3 will be the opposite of the cosine of negative pi over 3. so negative one half. If I add pi to 0, I get pi. The cosine of pi will be the opposite of the cosine of 0, -1. And you keep continuing in this fashion let me erase this so I can get have some space. Add pi, 4 pi over 3 take the opposite value. Negative one half. Add pi 3 pi over 2, take the opposite value 0. Add pi again 5 pi over 3. Take the opposite value one half, and we're almost done. Add pi to this and you get 2 pi and you've come through a full period. The opposite of this is 1.

So you can use this identity to extend the values of cosine or sine which is really handy for graphing.

So just a recap. We've got the co-function identities, sine of pi over 2 minus theta equals cosine theta. Cosine of pi over 2 minus theta equals sine theta and tangent of pi over 2 minus theta equals cotangent theta. These identities are actually true for all 6 trig functions. Then you've got the, let's call them the add pi identities first sine and cosine. If you add pi to theta, you get the opposite of sine theta. The cosine of theta plus pi minus cosine theta. And of course tangent of theta plus pi is tangent theta because tangent has period pi.