# How do you find the exact value of sin (x/2) and cos (x/2) if cos x= - 1/4 and pi< x<2pi

How do you find the exact value of $\mathrm{sin}\left(\frac{x}{2}\right)$ and $\mathrm{cos}\left(\frac{x}{2}\right)$ if $\mathrm{cos}x=-\frac{1}{4}$ and $\pi ?
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autarhie6i
Use the trig identity: $\mathrm{cos}x=1-2{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)$
$-\frac{1}{4}=1-2{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)\to {\mathrm{sin}}^{2}\left(\frac{x}{2}\right)=\frac{5}{8}=0.625$
$\mathrm{sin}x/2=-0.79$
Next, use trig identity: ${\mathrm{cos}}^{2}\left(\frac{x}{2}\right)-{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)=\mathrm{cos}x$
${\mathrm{cos}}^{2}\left(\frac{x}{2}\right)=-\frac{1}{4}+\frac{5}{8}=\frac{3}{8}\to \ast \frac{\mathrm{cos}x}{2}=-0.61\ast$
Check
${\mathrm{sin}}^{2}\left(\frac{x}{2}\right)+{\mathrm{cos}}^{2}\left(\frac{x}{2}\right)=\frac{5}{8}+\frac{3}{8}=\frac{8}{8}=1$