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Question Video: ο»ΏDetermining the Variation Function of the Cosine Function Mathematics

Determine the variation function of 𝑓(π‘₯) = cos π‘₯ at π‘₯ = πœ‹/2.

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Video Transcript

Determine the variation function of 𝑓 of π‘₯ is equal to the cos of π‘₯ at π‘₯ is equal to πœ‹ by two.

In this question, we’re asked to determine the variation function of a given trigonometric function: 𝑓 of π‘₯ is the cos of π‘₯. And we’re told to do this at a value of π‘₯ is equal to πœ‹ 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 π‘₯-inputs change from π‘Ž to π‘Ž plus β„Ž. In particular, the variation function 𝑣 of β„Ž of a function 𝑓 of π‘₯ at π‘₯ is equal to β„Ž is given by 𝑓 evaluated at π‘Ž plus β„Ž minus 𝑓 evaluated at π‘Ž. So, in our case, our function 𝑓 of π‘₯ is the cos of π‘₯ and our value of π‘Ž is πœ‹ by two.

So, if we substitute our function 𝑓 of π‘₯ is equal to the cos of π‘₯ and π‘Ž is equal to πœ‹ by two into this equation, we get 𝑣 of β„Ž is equal to the cos of πœ‹ by two plus β„Ž minus the cos of πœ‹ by two. This is one expression for the variation function 𝑣 of β„Ž.

However, we can simplify this expression. First, we know the cos of πœ‹ by two is equal to zero. This means 𝑣 of β„Ž is equal to the cos of πœ‹ by two plus β„Ž. 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œ‹ by two plus πœƒ is equal to negative the sin of πœƒ. Therefore, the cos of πœ‹ by two plus β„Ž is just equal to negative the sin of β„Ž. And we can’t simplify this any further.

So we were able to show the variation function of the function 𝑓 of π‘₯ is equal to the cos of π‘₯ at π‘₯ is equal to πœ‹ by two is 𝑣 of β„Ž is equal to negative the sin of β„Ž.

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