Question Video: Finding the Variation Function at a Point for a Given Trigonometric Function

Determine the variation function of 𝑓(π‘₯) = π‘Ž sin π‘₯ at π‘₯ = πœ‹. If 𝑉(πœ‹/2) = 1, find π‘Ž.

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Video Transcript

Determine the variation function of 𝑓 of π‘₯ is equal to π‘Ž sin π‘₯ at π‘₯ is equal to πœ‹. If 𝑉 of πœ‹ by two is equal to one, find π‘Ž.

There are two parts to this question. We’re asked to find the variation function for a trigonometric function 𝑓 of π‘₯ is π‘Ž sin π‘₯ at π‘₯ is equal to πœ‹ and then to find the value of π‘Ž using the fact that 𝑉 of πœ‹ by two is equal to one. Let’s begin by recalling that for a function 𝑓 of π‘₯, the variation function 𝑉 of β„Ž at π‘₯ is equal to 𝛼 is 𝑉 of β„Ž is 𝑓 of 𝛼 plus β„Ž minus 𝑓 of 𝛼. And that’s where β„Ž is the change in π‘₯. In our case, we’re given that π‘₯ is equal to πœ‹, and that’s equal to our 𝛼. This means that our variation function 𝑉 of β„Ž is 𝑓 of πœ‹ plus β„Ž minus 𝑓 of πœ‹.

We’re going to need to find then 𝑓 of πœ‹ plus β„Ž and 𝑓 of πœ‹. So first, substituting π‘₯ is equal to πœ‹ plus β„Ž into our function 𝑓, we have 𝑓 of πœ‹ plus β„Ž is equal to π‘Ž times the sin of πœ‹ plus β„Ž. 𝑓 of πœ‹ is simply π‘Ž sin πœ‹ so that our function 𝑉 of β„Ž is π‘Ž sin πœ‹ plus β„Ž minus π‘Ž sin πœ‹. In fact, we know that sin πœ‹ is equal to zero, so π‘Ž sin πœ‹ is zero. And we have 𝑉 of β„Ž is π‘Ž sin πœ‹ plus β„Ž. For our expression sin πœ‹ plus β„Ž, we can use the identity the sin of πœ‹ plus or minus capital 𝐴 is equal to negative or positive the sin of capital 𝐴. In our case, where the angle capital 𝐴 corresponds to β„Ž, we have negative lowercase π‘Ž times the sin of β„Ž. Our variation function 𝑉 of β„Ž is therefore negative π‘Ž sin β„Ž.

Next, we’re asked to find the value of π‘Ž if 𝑉 of πœ‹ by two is equal to one. We can do this by substituting β„Ž is equal to πœ‹ by two into 𝑉 of β„Ž and putting this equal to one. We then have 𝑉 of πœ‹ by two is equal to negative π‘Ž sin πœ‹ by two is equal to one. In fact, we know that sin πœ‹ by two is equal to one so that we have negative π‘Ž times one is equal to one. That is, negative π‘Ž is equal to one. And multiplying both sides by negative one, we have π‘Ž equal to negative one. The variation function of 𝑓 of π‘₯ is equal to π‘Ž sin π‘₯ at π‘₯ is equal to πœ‹ is 𝑉 of β„Ž is negative π‘Ž sin β„Ž. And if 𝑉 of πœ‹ by two is equal to one, then π‘Ž is equal to negative one.

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