How do you simplify $cos ({sin}^{- 1} (\frac{3}{5}) - {cos}^{- 1} (\frac{1}{2}))$ $cos(sin^-1(3/5)-cos^-1(1/2))$ ?

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How do you simplify $cos ({sin}^{- 1} (\frac{3}{5}) - {cos}^{- 1} (\frac{1}{2}))$ $cos(sin^-1(3/5)-cos^-1(1/2))$ ?

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Answer 1 · 2016-05-13T04:41:14

$(\frac{1}{10}) (\pm 4 \pm 3 \sqrt{3})$ $(1/10)(+-4+-3sqrt3)$

Explanation:

Let $a = {sin}^{- 1} (\frac{3}{5}) and b = {cos}^{- 1} (\frac{1}{2})$ $a = sin^(-1)(3/5) and b = cos ^(-1)(1/2)$ .

Then, $sin a = \frac{3}{5}, cos a = \pm \frac{4}{5}, cos b = \frac{1}{2} and sin b = \pm \frac{\sqrt{3}}{2}$ $sin a = 3/5, cos a = +-4/5, cos b = 1/2 and sin b = +-sqrt3 /2$

Now, the given expression is $cos (a - b)$ $cos (a - b )$

$= cos a cos b + sin a sin b$ $= cos a cos b + sin a sinb$

$= \pm \frac{2}{5} \pm 3 \frac{\sqrt{3}}{10} = (\frac{1}{10}) (\pm 4 \pm 3 \sqrt{3})$ $=+-2/5+-3sqrt3/10=(1/10)(+-4+-3sqrt3)$ .

The choice of sign depends on the specified ranges (if any ), for the inverse angles.a and b.

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How do you simplify $cos ({sin}^{- 1} (\frac{3}{5}) - {cos}^{- 1} (\frac{1}{2}))$ $cos(sin^-1(3/5)-cos^-1(1/2))$ ?

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