How do you evaluate ${csc}^{- 1} (- \sqrt{2})$ $csc^-1(-sqrt2)$ without a calculator?

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How do you evaluate ${csc}^{- 1} (- \sqrt{2})$ $csc^-1(-sqrt2)$ without a calculator?

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Answer 1 · 2018-06-21T04:29:21

$\frac{5 π}{4}; \frac{7 π}{4}$ $(5pi)/4; (7pi)/4$

Explanation:

$csc x = - \sqrt{2}$ $csc x = -sqrt2$ --> $sin x = \frac{1}{csc} = - \frac{1}{\sqrt{2}} = - \frac{\sqrt{2}}{2}$ $sin x = 1/(csc) = - 1/sqrt2 = - sqrt2/2$
${csc}^{- 1} (- \sqrt{2}) - \to arcsin (- \frac{\sqrt{2}}{2})$ $csc^-1(-sqrt2) --> arcsin (-sqrt2/2)$
Trig table and unit circle give 2 solutions:

$x = - \frac{π}{4}$ $x = -pi/4$ , or $x = \frac{7 π}{4}$ $x = (7pi)/4$ (co-terminal), and
$x = π - (- \frac{π}{4}) = \frac{5 π}{4}$ $x = pi - (-pi/4) = (5pi)/4$

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How do you evaluate ${csc}^{- 1} (- \sqrt{2})$ $csc^-1(-sqrt2)$ without a calculator?

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How do you evaluate csc−1(−√2)csc^-1(-sqrt2) without a calculator?

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How do you evaluate ${csc}^{- 1} (- \sqrt{2})$ $csc^-1(-sqrt2)$ without a calculator?