Reciprocal Trigonometric Functions on the Unit Circle

9. Unit Circle

Reciprocal Trigonometric Functions on the Unit Circle - Video Tutorials & Practice Problems

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Secant, Cosecant, & Cotangent on the Unit Circle

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Hey, everyone up to this point, we've been working with three trig functions sign cosine and tangent on our unit circle. But remember there are actually three other trig functions are reciprocal trig functions, cos and cotangent. Now, you may be worried that we're going to have to add a bunch of new information on our unit circle to find these reciprocal trig functions. But you don't have to worry about that at all because we can find the values of co can see it and cotangent using what's already on the unit circle. So let's go ahead and jump right in here. Now, these functions remember are called the reciprocal trig functions because they are simply one over our three original trig functions, sine, cosine and tangent. So for the cosecant of an angle, this is really just one over the sine of that angle. And on the unit circle, since we know that the s of an angle is equal to that Y value, the cosecant of the angle is just one over Y. Then for the sequent, since this is one over cosine of that angle, and we know that the cosine of any angle on our unit circle is just that X value, the sequent of that angle is one over X. So looking at our unit circle, if I asked you to find the C and cosecant of 60 degrees, you could simply take one over that X value and one over that Y value. And you're done. Now, let's not forget the tangent here because we also have the cotangent as one of our reciprocal trig functions. And when we find the cotangent of angle, this is just one over the tangent. Now we know the tangent of an angle to be Y over X on our unit circle. So we can simply flip that fraction to get the reciprocal telling us that the code tangent of an angle is simply X over Y. So again, if I asked you to find the code tangent of 60 degrees, you could go over to your unit circle, take your X value, divide it by your Y value and you'd be good to go. Now with that in mind, let's go ahead and work through some examples together. So our first example here is the cos of pi over six. Now we know that the cos is one over the sign of that angle. So this is one over the sign of pi over six, which we also know to be the Y value. So going up to our unit circle here at pi over six, looking at that Y value of one half here for the cos of pi over six, I can take 1/1 half. Now, whenever I take the reciprocal of a fraction, I'm really just flipping that fraction. So this gives me a value of two. So the cos of pi over six is equal to two. Now, let's look at the code tangent here. Now, here we're asked to find the cotangent of pi over four. Now, we know that the cotangent of any angle is just our X value over our Y value. So here we think about that again, X over Y. Now we can go up to our unit circle at pi over four, look at those X and Y values and simply divide them. So here we have the square root of 2/2 divided by the square root of 2/2. Now, because these are the exact same value, we end up with an answer of one. The cotangent of pi over four is just one, which just so happens to be the exact same as the tangent of pi over four because those X and Y values are the same. Now, let's look at one final example here here we have the C cat of zero. Now the C can of zero, I know that this is just one over the X value at zero degrees. So you can go up here to my unit circle at zero degrees, identify that X value and plug that in. So this is just going to be 1/1 which is of course, just equal to one. So the C can of zero is just one. And we're done here now that we know how to find these reciprocal trig values from all of the info we already have on the unit circle. Let's get some more practice. Thanks for watching and let me know if you have questions.