How do you evaluate $tan ({sin}^{- 1} (- \frac{5}{11}))$ $tan(sin^-1(-5/11))$ without a calculator?

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How do you evaluate $tan ({sin}^{- 1} (- \frac{5}{11}))$ $tan(sin^-1(-5/11))$ without a calculator?

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Answer 1 · 2016-10-11T15:50:26

$- - \frac{5}{24} \sqrt{6}$ $--5/24sqrt 6$ .

Explanation:

Let $a = {sin}^{- 1} (- \frac{5}{11}) \in Q_{4}$ $a = sin^(-1)(-5/11) in Q_4$ , for the principal value.

Then, $sin a = - \frac{5}{11}, cos a = \sqrt{1 - {(- \frac{5}{11})}^{2}} = \frac{4}{11} \sqrt{6}$ $sin a = -5/11, cos a = sqrt(1-(-5/11)^2)= 4/11sqrt6$

The given expression is

$tan a = \frac{sin a}{cos a} = - \frac{5}{4 \sqrt{6}} = - \frac{5}{24} \sqrt{6}$ $tan a = sin a/cos a= -5/(4sqrt 6)=-5/24sqrt 6$

Of course, the general value $a \in Q_{3}$ $a in Q_3$ also. wherein tana is

positive and cos a is negative,

So, ignoring the principal-value-convention for inverse sine, the

answer could be presented as

$= \pm \frac{5}{24} \sqrt{6}$ $=+-5/24sqrt6$ .

For me, this is the answer. You can choose yours.

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How do you evaluate $tan ({sin}^{- 1} (- \frac{5}{11}))$ $tan(sin^-1(-5/11))$ without a calculator?

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