Question #3960a

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Question #3960a

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Answer 1 · 2017-05-29T19:23:53

interval $(\frac{π}{4}, \frac{3 π}{4})$ $(pi/4, (3pi)/4)$

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 $π$ $pi$
a. sin x = 0--> x = 0 and $x = π$ $x = pi$ .
For $(0, π)$ $(0, pi)$ , the function f(x) = sin x > 0
b. cos 2x = 0 --> $2 x = \frac{π}{2}$ $2x = pi/2$ and $2 x = 3 \frac{π}{2}$ $2x = 3pi/2$ -->
$x = \frac{π}{4}$ $x = pi/4$ and $x = \frac{3 π}{4}$ $x = (3pi)/4$
Inside interval $(\frac{π}{4}, 3 \frac{π}{4})$ $(pi/4, 3pi/4)$ , the function g(x) = cos 2x < 0

Variation of f(x)
0 + + + + + + $\frac{π}{4}$ $pi/4$ + + + ++ + $\frac{π}{2}$ $pi/2$ + + + + + + $\frac{3 π}{4}$ $(3pi)/4$ + + ++ + + $π$ $pi$

Variation of g(x)
0+++++++++ $\frac{π}{4}$ $pi/4$ - - - - - - - - $\frac{π}{2}$ $pi/2$ - - - - - - - $\frac{3 π}{4}$ $(3pi)/4$ ++++++++ $π$ $pi$

The resultant F(x) = f(x).g(x) < 0 when x inside interval $(\frac{π}{4}, \frac{3 π}{4})$ $(pi/4, (3pi)/4)$

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Question #3960a

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