Question

List all x-intercepts for y=-3sin[(1/2)x+(1/5)pi] on the interval (-(2/5)pi,4pi)

Answer 1

I'm having a hard time figuring this one out.

Answer 2

We have
\[y=-3 \sin {(\frac 12 x +\frac 15 \pi})\]
we need to find x - intercepts
so y=0
\[0=-3 \sin {(\frac 12 x +\frac 15 \pi})\]
so it boils down to
\[0=\sin {(\frac 12 x +\frac 15 \pi})\]
do you know for what value of x sin x=0?

Answer 3

At 0 and pi, right?

Answer 4

these are just two values, it's 0 for n*pi
n=0, 1, 2, 3,....
let's put n=0, so we get
\[\frac 1 2 x+\frac 1 5 \pi=0\]
from this what do you get for x?

Answer 5

I'm stuck.

Answer 6

\[\frac 12 x=-\frac 1 5 \pi\]
\[x=-\frac 25 \pi\]
does this lie in our interval?

Answer 7

Yes, it does.

Answer 8

we have the interval as
\[(-\frac 2 5 \pi, 4\pi)\]
it's open interval the end limits won't be considered

Answer 9

All of the possible choices have three x-intercepts. And 4pi isn't one. How do I know how many increments to move along the x-axis before I reach 4pi?

Answer 10

now put n=1 so
\[\frac 1 2 x+\frac 1 5 \pi=\pi\]
now find x from this
we've to find x for n=2, 3 ,4 until x is greater than 4pi

Answer 11

Ah, I see. So (8/5)pi should be next then, right?

Answer 12

yeah, which does lie in the interval
then n=2
continue doing this and do check every time that x lies in the interval

Answer 13

Right, until it exceeds 4pi. I think I got it. Thanks, ash.

Answer 14

you're welcome :D