Lucia Grimes

2022-07-14

Prove: $\mathrm{sin}2\theta \phantom{\rule{thickmathspace}{0ex}}\ge \phantom{\rule{thickmathspace}{0ex}}\frac{1}{2}\left(-1-3{\mathrm{cos}}^{2}\theta \right)$

furniranizq

Expert

Hint: Since $1={\mathrm{sin}}^{2}\left(t\right)+{\mathrm{cos}}^{2}\left(t\right)$ and $2\mathrm{sin}\left(2t\right)=4\mathrm{sin}\left(t\right)\mathrm{cos}\left(t\right)$
$2\mathrm{sin}\left(2t\right)+1+3{\mathrm{cos}}^{2}\left(t\right)={\left(\mathrm{sin}\left(t\right)+2\mathrm{cos}\left(t\right)\right)}^{2}.$

Do you have a similar question?