### Video Transcript

Determine the variation function of
π of π₯ is equal to π sin π₯ at π₯ is equal to π. If π of π by two is equal to one,
find π.

There are two parts to this
question. Weβre asked to find the variation
function for a trigonometric function π of π₯ is π sin π₯ at π₯ is equal to π and
then to find the value of π using the fact that π of π by two is equal to
one. Letβs begin by recalling that for a
function π of π₯, the variation function π of β at π₯ is equal to πΌ is π of β is
π of πΌ plus β minus π of πΌ. And thatβs where β is the change in
π₯. In our case, weβre given that π₯ is
equal to π, and thatβs equal to our πΌ. This means that our variation
function π of β is π of π plus β minus π of π.

Weβre going to need to find then π
of π plus β and π of π. So first, substituting π₯ is equal
to π plus β into our function π, we have π of π plus β is equal to π times the
sin of π plus β. π of π is simply π sin π so
that our function π of β is π sin π plus β minus π sin π. In fact, we know that sin π is
equal to zero, so π sin π is zero. And we have π of β is π sin π
plus β. For our expression sin π plus β,
we can use the identity the sin of π plus or minus capital π΄ is equal to negative
or positive the sin of capital π΄. In our case, where the angle
capital π΄ corresponds to β, we have negative lowercase π times the sin of β. Our variation function π of β is
therefore negative π sin β.

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