Mabel Breault

2022-01-16

Use half-angle identities to find an exact value without a calculator.
${\mathrm{tan}195}^{\circ }$

vrangett

Expert

Let . We can reduce u to ${30}^{\circ }$ by subtracting ${360}^{\circ }$. Now use one of the tan half angle identities
$\mathrm{tan}\frac{u}{2}=\frac{1-\mathrm{cos}u}{\mathrm{sin}u}$
${\mathrm{tan}195}^{\circ }=\frac{1-{\mathrm{cos}30}^{\circ }}{{\mathrm{sin}30}^{\circ }}=\frac{1-\frac{\sqrt{3}}{2}}{\frac{1}{2}}=2-\sqrt{3}$
Result:
$2-\sqrt{3}$

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