Question

a) \(2\cos^2-1=0\)
b) -2tan2x + 3 = 0
c) \(\sqrt{3}.\cot\left(2x+\dfrac{\Pi}{4}\right)-1=0\)
d) \(\dfrac{1}{2}.\sin\left(x-\dfrac{\Pi}{3}\right)-\dfrac{1}{4}=0\)
e) \(2\cos^2x-3\cos x+1=0\)

Answer 1

`a)2cos^2 x-1=0`

`<=>1+cos 2x-1=0`

`<=>cos 2x=0<=>2x=\pi/2+k\pi<=>x=\pi/4+k\pi/2`   `(k in ZZ)`

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`b)-2tan 2x+3=0`

`<=>tan 2x=3/2`

<=>2x=arctan(3/2)+k\pi<=>x=1/2arcta(3/2)+k\pi/2`   `(k in ZZ)`

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`c)\sqrt{3}cot(2x+\pi/4)-1=0`

`<=>cot(2x+\pi/4)=\sqrt{3}/3`

`<=>2x+\pi/4=\pi/3+k\pi`

`<=>x=\pi/24+k\pi/2`   `(k in ZZ)`

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`d)1/2sin(x-\pi/3)-1/4=0`

`<=>sin(x-\pi/3)=1/2`

`<=>[(x-\pi/3=\pi/6+k2\pi),(x-\pi/3=[5\pi]/6+k2\pi):}`

`<=>[(x=\pi/2+k2\pi),(x=[7\pi]/6+k2\pi):}`   `(k in ZZ)`

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`e)2cos^2 x-3cos x+1=0`

`<=>[(cos x=1),(cos x=1/2):}`

`<=>[(x=k2\pi),(x=+-\pi/3+k2\pi):}`    `(k in ZZ)`