Question #1a1ed

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Question #1a1ed

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Answer 1 · 2016-10-25T06:05:14

$x = \frac{\sqrt{3}}{3}$ $x=sqrt(3)/3$

Explanation:

$arccot (x) + 2 arcsin (\frac{\sqrt{3}}{2}) = π$ $"arccot"(x) + 2arcsin(sqrt(3)/2) = pi$

The inverse sine function $arcsin (x)$ $arcsin(x)$ is defined as the unique value in the interval $[- \frac{π}{2}, \frac{π}{2}]$ $[-pi/2,pi/2]$ such that $sin (arcsin (x)) = x$ $sin(arcsin(x)) = x$ . On that interval, we have $sin (\frac{π}{3}) = \frac{\sqrt{3}}{2}$ $sin(pi/3) = sqrt(3)/2$ as a well known angle. Thus $arcsin (\frac{\sqrt{3}}{2}) = \frac{π}{3}$ $arcsin(sqrt(3)/2) = pi/3$

$\Rightarrow arccot (x) + \frac{2 π}{3} = π$ $=> "arccot"(x) + (2pi)/3 = pi$

$\Rightarrow arccot (x) = \frac{π}{3}$ $=> "arccot"(x) = pi/3$

$\Rightarrow cot (arccot (x)) = cot (\frac{π}{3})$ $=> cot("arccot"(x)) = cot((pi)/3)$

$\Rightarrow x = cot (\frac{π}{3})$ $=> x = cot((pi)/3)$

$= \frac{cos (\frac{π}{3})}{sin (\frac{π}{3})}$ $=cos((pi)/3)/sin((pi)/3)$

$= \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$ $=(1/2)/(sqrt(3)/2)$

$= \frac{1}{\sqrt{3}}$ $=1/sqrt(3)$

$= \frac{\sqrt{3}}{3}$ $=sqrt(3)/3$

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