How do you find the interval of decrease and increase of a sin function? Given this function f(x) = sin2x-90degrees  What is the interval of increase and decrease and how do you find the zeros?...

How do you find the interval of decrease and increase of a sin function?

Given this function

f(x) = sin2x-90degrees

 

What is the interval of increase and decrease and how do you find the zeros?

Your help is greatly appreciated! Thanks

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neela | High School Teacher | (Level 3) Valedictorian

Posted on

f(x)=sin2x -90

The function f(x) is incresing when f'(x) is >0.

f'(x) = (sin2x -90)' = 2cos2x

2co2x > 0 implies cos2x > 0

Cos2x > 0 when  2npi-pi/2 < 2x < 2npi-pi.

Cos2x > 0 when  -pi/4 < x <pi/4. That is when x is in (npi/2-pi/4 , npi/2+pi/4),  n = 0,1,2,..

Therefore sin2x - 90 is increasing in (npi-pi/4 ,npi+ pi/4).

 Or

When x  is   in the interval (180n-45 deg ,180n+45 deg ) for n =0,1,2,....

Sin2x -90 is decreasing  when (sin2x-90)' < 0. Or

2cos2x < 0 or cos2x < 0.

cos2x < 0 when 2npi+ pi/2 < 2x < 2npi+3pi/2.

Or

npi+Pi/4 < x < npi+3pi/4 is  same as when x is in (180n+45deg to 135 deg.), n =0,1,2,3....

Therefore sin2x-90 is decreasing in (180n+45 deg , 180n+135deg), for n = 0,1,2,3....

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