Question

Theta/2=arcsin(1/2)

1. On the interval [0,2pi] what values of theta/2 satisfy the equation?
2. On the interval [0,2pi] what values of theta does your answer to part 2 produce?
3. Write an expression for all solutions to the equation.

Answer 1

`\bf \cfrac{\theta }{2}=sin^(-1)\left( (1)/(2) \right)\qquad \textit{let's say that }sin^(-1)\left( (1)/(2) \right)=\alpha\\\\\\\cfrac{\theta }{2}=\alpha\implies \theta =2\alpha\\\\-------------------------------`

`\bf sin^(-1)\left( (1)/(2) \right)=\alpha\implies sin(\alpha)=\cfrac{1}{2}\implies \alpha=\begin{cases}(\pi )/(6)\\\\(5\pi )/(6)\end{cases}\\\\\\\theta =2\alpha\implies \theta =\begin{cases}2\cdot (\pi )/(6)\implies (\pi )/(3)\\\\2\cdot (5\pi )/(6)\implies (5\pi )/(3)\end{cases}\\\\\\\stackrel{\textit{all solutions for }\theta }{\pm\cfrac{\pi }{3}\pm\cfrac{4\pi n}{3}}\qquad \{n~|~ n\in \mathbb{Z}\}`