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I have a question that requires the use of optimisation to solve for the following expression:

$$\cos ec{(\cos^{-1}{(-\frac{\sqrt{3}}{2})}+\sin^{-1}{(-\frac{\sqrt{3}}{2})})}$$

I'm a bit baffled, as I'm not sure whether this refers to finding the value of cosec or finding the value of cos(ec)(cos^(-1)...). No further clarity is given.

Without using optimisation, the expression can just be evaluated at face-value to:

$$\csc{(\frac{5\pi}{6}+\frac{4\pi}{3})} = \csc{(2\pi+\frac{\pi}{6})} = \frac{1}{\sin{\frac{\pi}{6}}} = \frac{1}{\frac{1}{2}} = 2$$

Any suggestions would be appreciated!

thisisanon
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  • What do you mean by optimisation in this context exactly? – danimal Mar 25 '15 at 12:14
  • In the course, we're using optimisation techniques like first and second derivative tests to determine minimum and maximum values. The worksheet specifies that optimisation should be used to determine the 'exact value' of the expression, but frankly I just don't understand how it applies in this context. – thisisanon Mar 25 '15 at 12:17
  • What is meant by the notation $\cos ec$? – Travis Willse Mar 25 '15 at 12:19
  • I have no idea. I've reproduced it exactly as it was printed in the worksheet. I can't see that it refers to cos(e*c), but I also don't know why the authors wouldn't just use csc instead of cosec if that's what they intended. – thisisanon Mar 25 '15 at 12:27

2 Answers2

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Do know you $$\arccos x+\arcsin x=\frac\pi2$$ for $-1\le x\le1$

Proof:Why it's true? $\arcsin(x) +\arccos(x) = \frac{\pi}{2}$

Community
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lab bhattacharjee
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Since $\left|-\dfrac{\sqrt{3}}{2}\right|\le 1$, we have that $\cos^{-1}\left( -\dfrac{\sqrt{3}}{2}\right)+\sin^{-1}\left( -\dfrac{\sqrt{3}}{2}\right) =\dfrac{\pi}{2} $.

So $\operatorname{cosec} \left( \cos^{-1}\left( -\dfrac{\sqrt{3}}{2}\right)+\sin^{-1}\left( -\dfrac{\sqrt{3}}{2}\right)\right)=\operatorname{cosec} \left(\dfrac{\pi}{2}\right) =\dfrac{1}{\sin \left(\dfrac{\pi}{2}\right)}=1 $.

ASB
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