How do you evaluate $sin(cos^-1(1/2))$ without a calculator?

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Answer 1 · 2018-03-05T14:59:40

$sin(cos^(-1)(1/2))=sqrt(3)/2$

Let $cos^(-1)(1/2)=x$ then $cosx=1/2$

$rarrsinx=sqrt(1-cos^2x)=sqrt(1-(1/2)^2)=sqrt(3)/2$

$rarrx=sin^(-1)(sqrt(3)/2)=cos^(-1)(1/2)$

Now, $sin(cos^(-1)(1/2))=sin(sin^(-1)(sqrt(3)/2))=sqrt(3)/2$

Answer 2 · 2018-03-05T15:13:41

$sin cos ^-1 (1/2)) = sqrt 3 / 2$

To find value of $sin (cos ^-1 (1/2))$

Let theta = cos^-1 (1/2)#

$cos theta = (1/2)$

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We know, from the above table, $cos 60 = 1/2$

Hence theta = 60^@#

Replacing $cos^-1 (1/2)$ with $theta = 60^@$ ,

The sum becomes, $=> sin theta = sin 60 = sqrt3 / 2$ (As per table above)

How do you evaluate sin(cos^-1(1/2))sin(cos^-1(1/2)) without a calculator?