Question

Use an identity to solve the following equation on the interval [0,21). 2 cos2x- sin x-1=0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. X= (Type your answer in radians. Use integers or fractions for any numbers in the expression. Type an exact answer, using a as needed. Use a comma to separate answers as needed.) OB. There is no solution.

Answer 1

2cos? - sint -1=0

$\small 2(1-sin^{2}x)-sinx-1=0$

$\small 2-2sin^{2}x-sinx-1=0$

$\small 2sin^{2}x+sinx-1=0$

2 sin’t + 2 sint - sint -1=0

$\small 2sinx(sinx+1)-1(sinx+1)=0$

$\small (2sinx-1)(sinx+1)=0$

$\small sinx=\frac{1}{2},-1$

$\small \textup{In interval }\left [ 0,2\pi \right ]$,

$\small \textup{i) }sinx=\frac{1}{2}\Rightarrow x=\frac{\pi }{6},\frac{5\pi }{6}$

$\small \textup{ii) }sinx=-1\Rightarrow x=\frac{3\pi }{2}$

$\small \boldsymbol{\therefore x=\left \{ \frac{\pi }{6},\frac{5\pi }{6},\frac{3\pi }{2} \right \}}$

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