Kai Kerr

2023-03-09

How to find the exact values of $\mathrm{sin}\left(\frac{u}{2}\right),\mathrm{cos}\left(\frac{u}{2}\right),\mathrm{tan}\left(\frac{u}{2}\right)$ using the half angle formulas given $\mathrm{sin}u=\frac{5}{13},\frac{\pi }{2}?

planeiaxuf

$\mathrm{sin}u=\frac{5}{13}$. First, find cos u.
${\mathrm{cos}}^{2}u=1-{\mathrm{sin}}^{2}u=1-\frac{25}{169}=\frac{144}{169}$ --> $\mathrm{cos}u=±\frac{12}{13}$.
Since u is in Quadrant 2, then, cos u < 0.
$\mathrm{cos}u=-\frac{12}{13}$.
To find $\mathrm{cos}\left(\frac{u}{2}\right)$, trigonometric identity in use
$2{\mathrm{cos}}^{2}\left(\frac{u}{2}\right)=1+\mathrm{cos}u=1-\frac{12}{13}=\frac{1}{13}$
${\mathrm{cos}}^{2}\left(\frac{u}{2}\right)=\frac{1}{26}$ --> $\mathrm{cos}\left(\frac{u}{2}\right)=±\frac{1}{\sqrt{26}}$.
Sin u is in Quadrant 2, then $\frac{u}{2}$ is in Quadrant 1, and $\mathrm{cos}\left(\frac{u}{2}\right)$ is positive:
$\mathrm{cos}\left(\frac{u}{2}\right)=\frac{1}{\sqrt{26}}=\frac{\sqrt{26}}{26}$
To find $\mathrm{sin}\left(\frac{u}{2}\right)$, apply the trig identity:
$\mathrm{sin}u=2\mathrm{sin}\left(\frac{u}{2}\right).\mathrm{cos}\left(\frac{u}{2}\right)$
$\mathrm{sin}\left(\frac{u}{2}\right)=\frac{\mathrm{sin}u}{2\mathrm{cos}\left(\frac{u}{2}\right)}=\left(\frac{5}{13}\right)\left(\frac{\sqrt{26}}{2}\right)=\frac{5\sqrt{26}}{26}$
$\mathrm{tan}\left(\frac{u}{2}\right)=\frac{\mathrm{sin}\left(\frac{u}{2}\right)}{\mathrm{cos}\left(\frac{u}{2}\right)}=\left(\frac{5\sqrt{26}}{26}\right)\left(\sqrt{26}\right)=5$

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