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

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Answer 1 · 2016-11-30T10:40:03

${sin}^{- 1} (- \frac{\sqrt{2}}{2}) = - \frac{π}{4}$ $sin^(-1)(-sqrt(2)/2) = -pi/4$

This is a known value for $sin (x)$ $sin(x)$ , you should remember that:

$sin (\frac{π}{4}) = \frac{\sqrt{2}}{2}$ $sin(pi/4) = sqrt(2)/2$

and that:

$sin (- x) = - sin (x)$ $sin(-x) = -sin(x)$

Although many other angles have the same value for sine, namely:

$α = - \frac{π}{4} + 2 k π$ $alpha = -pi/4+2kpi$
and
$α = - \frac{3 π}{4} + 2 k π$ $alpha = -(3pi)/4 +2kpi$

conventionally $arcsin (x)$ $arcsin(x)$ is defined in the interval $[- \frac{π}{2}, \frac{π}{2}]$ $[-pi/2,pi/2]$
so you pick the value:

$x = - \frac{π}{4}$ $x=-pi/4$

How do you evaluate sin−1(−√22)sin^-1(-sqrt2/2) without a calculator?