MHB Inverse trigonometric functions

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The discussion centers on evaluating the expression $1. ~ \arccos(\cos\frac{4\pi}{3})$. Participants clarify that since the range of the arccosine function is $[0, \pi]$, $\cos\frac{4\pi}{3}$ should be expressed in that range. The correct transformation shows that $\cos\frac{4\pi}{3} = \cos\frac{2\pi}{3}$, leading to the conclusion that $\arccos(\cos\frac{4\pi}{3}) = \frac{2\pi}{3}$. A typographical error in the calculations is noted, indicating a possible confusion in the values used. The final consensus confirms that the answer is indeed $\frac{2\pi}{3}$.
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What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct?

The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$

$\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$
 
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There's an error in your last line.

Using symmetry,

$\cos\dfrac{4\pi}{3}=\cos\left(\pi-\dfrac{\pi}{3}\right)=\cos\dfrac{2\pi}{3}$
 
greg1313 said:
There's an error in your last line.

Using symmetry,

$\cos\dfrac{4\pi}{3}=\cos\left(\pi-\dfrac{\pi}{3}\right)=\cos\dfrac{2\pi}{3}$

Thanks. My reasoning was that cosine has a period $2\pi$. Where have I messed up?
 
Guest said:
What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct?

The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$

$\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$

$\displaystyle \begin{align*} \arccos{ \left[ \cos{ \left( \frac{4\,\pi}{3} \right) } \right] } &= \arccos{ \left[ \cos{ \left( \pi + \frac{\pi}{3} \right) } \right] } \\ &= \arccos{ \left[ -\cos{ \left( \frac{\pi}{3} \right) } \right] } \\ &= \arccos{ \left( -\frac{1}{2} \right) } \\ &= \pi - \arccos{ \left( \frac{1}{2} \right) } \\ &= \pi - \frac{\pi}{3} \\ &= \frac{2\,\pi}{3} \end{align*}$
 
Guest said:
My reasoning was that cosine has a period $2\pi$. Where have I messed up?

You typed a '3' where you probably intended a '4'.
 
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