How to Divide a Trig Function

Dividing by a trig function is something that you have to do in a very specific way. Find out how to divide by a trig function with help from an experienced math tutor in this free video clip.

Part of the Video Series: High School Math

Video Transcript

Hi there. This is Ryan Malloy here at the Worldwide Center of Mathematics. In this video, we're going to discuss how to divide a trig function. So suppose that we are given some expression like tangent of theta over the cosine of theta and we want to try to figure out how to write this in a number of different forms. Let's say we want to try to rewrite this in such a way that all of the terms are in the numerator and there's simply one in the denominator or perhaps we want to write it in such a way that there is a one in the numerator and all of the rest of the information is in the denominator. Or perhaps we want to write it in such a way that we only use the basic sin and cosine functions. There's many different ways we can manipulate it from here but before we do that let's go over a few definitions. Call that we have our two basic trig functions, sin and cosine and all other trig functions are defined based on these. Where tangent is equal to sin over cosine. We have cotangent which is equal to cosine over sin. We have secant which is equal to one over cosine and finally we have cosecant which is equal to one over sin. Well using these definitions we can manipulate this any way we want to. So, let's work this way. We can see that we rewrite tangent of theta as sin over cosine, what we get is sin theta over cosine theta times this denominator here, one over cosine theta. So one thing that we could write this as is sin theta over cosine squared theta. So this would be an acceptable answer if you were trying to write this in a form that only uses the two basic functions of sin and cosine. But let's say that we want to write this with only terms in the numerator and a one in the denominator like we said earlier. Okay, well again, we use the same formula here but let's take this one over cosine theta and recall that is equal to secant. So we get secant theta times tangent theta. That works just fine. We could rewrite it a different way if we didn't want to use tangent theta at all. We get this sin, this becomes secant and this becomes secant so this becomes sin theta, secant squared theta. Alright, let's do one more. How can we write this in such a way that all of our interesting terms are in the denominator and we just have one in the numerator? Well we'll keep our cosine in there and we recall that since tangent is equal to sin over cosine we'll end up having a cosine squared in the denominator but sin, we look at this formula here, cosecant of theta is equal to one over sin theta but the inverse is also true. We can write sin theta as one over the cosecant. So we'll extend this out a little bit farther and there we have it, all four of these terms are equivalent to our original function of tangent theta over cosine theta. My name is Ryan Malloy and we've just discussed how to divide a trig function.

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