How do you find the intervals of increase and decrease for
r(x) = sin x
The answer at the back of my textbook was (90( 4k - 1), 90 (4k + 1)
for the increase and (90(4k+1), 90(4k + 3) for the interval of decrease but I don't understand how they got that. Please explain
For the function r(x)=sin x.
In the first quadrant i.e. 0 to 90 degree: the value of sin x increases.
In the second quadrant i.e. 90 to 180 degrees: the value of sin x decreases.
In the third quadrant i.e. 180 to 270 degrees: the value of sin x decreases.
In the fourth quadrant i.e. 270 to 360 degrees: the value of sin x increases.
As the sequence of increase and decrease repeats itself once 360 degrees has been reached, it has been explained in your book in that format.
As you may see the interval of increase is (90( 4k - 1), 90 (4k + 1), substitute k=0, you get -90 to 90, this corresponds to the 1st and 4th quadrants. The same will apply for any value of k that you take.
The interval of decrease is (90(4k+1), 90(4k + 3), substitute k=0, you get 90 to 270, this corresponds to the 2nd and 3rd quadrants. The same will apply for any other value of k.
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.