Question Video: Determining the Variation Function of the Cosine Function Mathematics

Video Transcript

Determine the variation function of 𝑓 of 𝑥 is equal to the cos of 𝑥 at 𝑥 is equal to 𝜋 by two.

In this question, we’re asked to determine the variation function of a given trigonometric function: 𝑓 of 𝑥 is the cos of 𝑥. And we’re told to do this at a value of 𝑥 is equal to 𝜋 by two. And to do this, let’s start by recalling what we mean by the variation function of a given function as a given value.

We can recall that the variation function gives us a measure of how a function changes when its 𝑥-inputs change from 𝑎 to 𝑎 plus ℎ. In particular, the variation function 𝑣 of ℎ of a function 𝑓 of 𝑥 at 𝑥 is equal to ℎ is given by 𝑓 evaluated at 𝑎 plus ℎ minus 𝑓 evaluated at 𝑎. So, in our case, our function 𝑓 of 𝑥 is the cos of 𝑥 and our value of 𝑎 is 𝜋 by two.

So, if we substitute our function 𝑓 of 𝑥 is equal to the cos of 𝑥 and 𝑎 is equal to 𝜋 by two into this equation, we get 𝑣 of ℎ is equal to the cos of 𝜋 by two plus ℎ minus the cos of 𝜋 by two. This is one expression for the variation function 𝑣 of ℎ.

However, we can simplify this expression. First, we know the cos of 𝜋 by two is equal to zero. This means 𝑣 of ℎ is equal to the cos of 𝜋 by two plus ℎ. We could leave our answer like this. However, we can simplify this even further. We do this by recalling one of the cofunction identities. The cos of 𝜋 by two plus 𝜃 is equal to negative the sin of 𝜃. Therefore, the cos of 𝜋 by two plus ℎ is just equal to negative the sin of ℎ. And we can’t simplify this any further.

So we were able to show the variation function of the function 𝑓 of 𝑥 is equal to the cos of 𝑥 at 𝑥 is equal to 𝜋 by two is 𝑣 of ℎ is equal to negative the sin of ℎ.