How do you find the points where the graph of the function # f(x)=sin2x+sin^2x# has horizontal tangents?

1 Answer
Mar 15, 2016

Horizontal tangent means neither increasing nor decreasing. Specifically, the derivative of the function has to be zero #f'(x)=0#.

Explanation:

#f(x)=sin(2x)+sin^2x#

#f'(x)=cos(2x)(2x)'+2sinx*(sinx)'#

#f'(x)=2cos(2x)+2sinxcosx#

Set #f'(x)=0#

#0=2cos(2x)+2sinxcosx#

#2sinxcosx=-2cos(2x)#

#sin(2x)=-2cos(2x)#

#sin(2x)/cos(2x)=-2#

#tan(2x)=-2#

#2x=arctan(2)#

#x=(arctan(2))/2#

#x=0.5536#

This is one point. Since solution was given out by #tan# , other points will be every π times the factor in #2x# meaning #2π#. So the points will be:

#x=0.5536+2n*π#

Where #n# is any integer.
graph{sin(2x)+(sinx)^2 [-10, 10, -5, 5]}