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

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

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Answer 1 · 2016-08-19T07:11:18

$tan ({csc}^{- 1} (\frac{\sqrt{7}}{2})) = \frac{1}{\sqrt{3}}$ $tan(csc^-1(sqrt(7)/2))=1/sqrt3$

Explanation:

Let $θ = {csc}^{- 1} (- 2)$ $theta=csc^-1(-2)$

or

$csc θ = - 2$ $csctheta=-2$

or

$\frac{1}{sin θ} = - 2$ $1/sintheta=-2$

or

$sin θ = - \frac{1}{2}$ $sintheta=-1/2$

or

$θ = {sin}^{- 1} (- \frac{1}{2})$ $theta=sin^-1(-1/2)$

Hence

$\frac{p}{h} = \frac{1}{2}$ $p/h=1/2$ (Since $θ$ $theta$ lies in third quadrant; hence $-$ $-$ sign not taken in consideration)

$cos θ = \frac{b}{h}$ $costheta=b/h$

or

$cos θ = \frac{\sqrt{h^{2} - p^{2}}}{h}$ $costheta=sqrt(h^2-p^2)/h$

or

$cos θ = \frac{\sqrt{(2^{2} - 1^{2})}}{2}$ $costheta=sqrt((2^2-1^2))/2$

or

$cos θ = \frac{\sqrt{4 - 1}}{2}$ $costheta=sqrt(4-1)/2$

or

$cos θ = \frac{\sqrt{3}}{2}$ $costheta=sqrt3/2$

Therefore
$tan θ = \frac{sin θ}{cos θ}$ $tantheta=sintheta/costheta$

or

$tan θ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$ $tan theta=(1/2)/((sqrt3)/2$

or

$tan θ = \frac{1}{\sqrt{3}}$ $tan theta=1/sqrt3$

or

$tan ({csc}^{- 1} (\frac{\sqrt{7}}{2})) = \frac{1}{\sqrt{3}}$ $tan(csc^-1(sqrt(7)/2))=1/sqrt3$

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

1 Answer

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