Question

Please help me solve this,

Use trigonometric identities to solve 2cos^2(θ)=sin(θ)+1 exactly for 0≤θ<2π. If there is more than one answer, enter your answers as a comma separated list.

Answer 1

`\cos^2\theta=1-\sin^2\theta`


`2\cos^2\theta=\sin\theta+1`

`2-2\sin^2\theta=\sin\theta+1`

`2\sin^2\theta+\sin\theta-1=0`

`(2\sin\theta-1)(\sin\theta+1)=0`


`2\sin\theta-1=0\implies \sin\theta=\frac12\implies\theta=\frac\pi6,\frac{5\pi}6`

`\sin\theta+1=0\implies\sin\theta=-1\implies\theta=\frac{3\pi}2`