Question

Answer 1

`a)\sqrt{3}sin x+cos x=1`

`<=>\sqrt{3}/2sin x+1/2cos x=1/2`

`<=>sin(x+\pi/6)=1/2`

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

`<=>[(x=k2\pi),(x=[2\pi]/3+k2\pi):}`         `(k in ZZ)`

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`b)sin 2x+cos 2x=\sqrt{2}sin x`

`<=>1/\sqrt{2}sin 2x+1/\sqrt{2}cos 2x=sin x`

`<=>sin(2x+\pi/4)=sin x`

`<=>[(2x+\pi/4=x+k2\pi),(2x+\pi/2=\pi-x+k2\pi):}`

`<=>[(x=-\pi/4+k2\pi),(x=\pi/6+k[2\pi]/3):}`   `(k in ZZ)`