How do you evaluate ${csc}^{- 1} (\frac{2}{\sqrt{3}})$ $csc^-1(2/sqrt3)$ without a calculator?

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How do you evaluate ${csc}^{- 1} (\frac{2}{\sqrt{3}})$ $csc^-1(2/sqrt3)$ without a calculator?

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Answer 1 · 2016-08-09T16:26:15

$\frac{π}{3} + 2 k π$ $pi/3 + 2kpi$
$2 \frac{π}{3} + 2 k π$ $2pi/3 + 2kpi$

Explanation:

Use trig table of special arcs:
$csc x = \frac{1}{cos x} = \frac{2}{\sqrt{3}}$ $csc x = 1/(cos x) = 2/sqrt3$
$sin x = \frac{\sqrt{3}}{2}$ $sin x = sqrt3/2$
Trig table and unit circle give 2 solutions for arc x
$x = \frac{π}{3}$ $x = pi/3$ and $x = π - \frac{π}{3} = \frac{2 π}{3}$ $x = pi - pi/3 = (2pi)/3$

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How do you evaluate ${csc}^{- 1} (\frac{2}{\sqrt{3}})$ $csc^-1(2/sqrt3)$ without a calculator?

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