How do you find the intervals of increase and decrease for r(x) = sin xThe answer at the back of my textbook was (90( 4k - 1), 90 (4k + 1) for...

Consider the interval 90(k-1) to 90(k+1) degrees. Let us
give k= 0.

Then the interval is 90(0-1)degrees to 90(0+1)
degrees. Or

the interval is -90 degrees to +90 degrees. The
function r(x) sinx is increasing in this interval:

sin(-90)
= -1, sin (-60) =  -sqrt3/2, sin (150) = -1/2,  sin (0) =
0

sin (30) = 1/2, sin (60) sert3/2 , sin (90) =
1.

So in the interval -90 degree to 90 degree or in the
interval (-90 , 90) sin x is a contnuously
incresing.

Similarly in the interval (90(4k-1) , 90(4k+1) )
, k = 0,1,2,3..., sinx , being perodic after every 360 degree degree behavves exacly
similaly increasing.

Like that in the interval (90 degree
to 270 degree) or (90(4k+1 , to 90(4k+3) ) k = 0,1,2,3..., r(x) is decreasing
continuously.

Hope this helps.