What is the value of ${cos}^{- 1} (cos (- \frac{π}{3}))$ $cos^-1(cos(-pi/3))$ ?

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Answer 1 · 2016-07-23T17:35:08

${cos}^{- 1} (cos (- \frac{π}{3})) = \frac{π}{3}$ $cos^-1(cos(-pi/3))=pi/3$ ..

We have, $costheta=cos(-theta), AA theta in RR.............(1)$ .

Next let us recall the following defn. of the $cos^-1$ function :-

$cos^-1x=theta, |x|<= 1 iff costheta=x, theta in [0,pi]...............(2)$

Now, using $(2)$ in $larr$ direction, keeping in mind that $costheta <=1$ ,

we have, $cos^-1(costheta)=theta, if, theta in [0,pi]...................(2')$

Now, $cos^-1(cos(-pi/3))=cos^-1(cos(pi/3))................[by (1)$ ],

&, here, since $pi/3 in [0,pi]$ , we get, by $(2')$ ,

$cos^-1(cos(-pi/3))=cos^-1(cos(pi/3))=pi/3$ ..

What is the value of cos^-1(cos(-pi/3))cos−1(cos(−π3))cos^-1(cos(-pi/3))?