Question #3960a

1 Answer
Nghi N
May 29, 2017

interval #(pi/4, (3pi)/4)#

Explanation:

Solve this trig inequality by the sign chart.
First solve sin x.cos 2x = 0
Either factor should be zero.
Consider the function F(x) = f(x).g(x) = (sin x)(cos 2x)
The common period of F(x ) is #pi#
a. sin x = 0--> x = 0 and #x = pi#.
For #(0, pi)#, the function f(x) = sin x > 0
b. cos 2x = 0 --> #2x = pi/2# and #2x = 3pi/2# -->
#x = pi/4# and #x = (3pi)/4#
Inside interval #(pi/4, 3pi/4)#, the function g(x) = cos 2x < 0

Variation of f(x)
0 + + + + + + #pi/4#+ + + ++ +#pi/2#+ + + + + +#(3pi)/4#+ + ++ + + #pi#

Variation of g(x)
0+++++++++#pi/4# - - - - - - - - #pi/2# - - - - - - - #(3pi)/4#++++++++#pi#

The resultant F(x) = f(x).g(x) < 0 when x inside interval #(pi/4, (3pi)/4)#

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