Question

Find every angle theta with \(\displaystyle{0}\le\theta\le{2}\pi{r}{a}{d}{i}{a}{n}{s},{\quad\text{and}\quad}{2}{{\sin}^{{2}}{\left(\theta\right)}}+{\cos{{\left(\theta\right)}}}={2}\)

Answer 1

\(\displaystyle{{\sin}^{{2}}=}{1}-{{\cos}^{{2}},}\)
so using x as \(\displaystyle\theta,{2}-{2}{{\cos}^{{2}}{x}}+{\cos{{x}}}={2},\)
\(\displaystyle{2}{{\cos}^{{2}}{x}}-{\cos{{x}}}={0},{\cos{{x}}}{\left({2}{\cos{{x}}}-{1}\right)}={0},\)
\(\displaystyle{\cos{{x}}}={0}{\quad\text{or}\quad}{0.5}\)
\(\displaystyle{x}=\frac{{\pi}}{{3}}{\quad\text{or}\quad}{60}°,\frac{{\pi}}{{2}}{\quad\text{or}\quad}{90}°,{3}\frac{{\pi}}{{2}}{\quad\text{or}\quad}{270}°,{5}\frac{{\pi}}{{3}}{\quad\text{or}\quad}{300}°.\)