Question 1201900
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Evaluate:  {{{arccos(cos(-4pi/3))}}}
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<pre>
The angle  {{{-4pi/3}}}  lies in the second quadrant, QII.


The cosine function of this angle has a negative value.


They want you find  the angle,  {{{arccos(cos(-4pi/3))}}},  which has the same value of cosine as the angle  {{{-4pi/3)}}}.


arccos is, BY THE DEFINITION,  an angle in the range from 0 to {{{pi}}},  [0,{{{pi}}}).


So, they want you find an angle in the range  [0,{{{pi}}}),  which has the same cosine value,
as  {{{cos(-4pi/3)}}}.  It is the angle in the range  [0,{{{pi}}}),  which has the same projection on x-axis
as the angle  {{{-4pi/3}}}.  The desired angle is,  {{{highlight(OBVIOUSLY)}}},   {{{2pi/3}}}.


Geometrically, it is the same angle as  {{{-4pi/3}}},  but this  {{{2pi/3}}}  is expressed and presented in the right range.


This  EXACT  value is  PRECISELY  what they want from you.  There is  NO  NEED  to transform it to numerical value in radians.
You may transform it in radians for the sake of your curiosity, if you want, but in reality, it is  NOT  NEEDED.


This exact value,  {{{2pi/3}}},  is  PRECISELY  what they want from you.


<U>ANSWER</U>.  {{{arccos(cos(-4pi/3))}}} = {{{2pi/3}}}.
</pre>

Solved, with all necessary explanations.


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In solution of this problem,  the major step is to get a precise understanding 
what they want from you, &nbsp;and formulate it clearly, &nbsp;as I did it in my post.


To understand my solution better, &nbsp;read and re-read it several times: &nbsp;as many times,
as you need to get &nbsp;FULL &nbsp;UNDERSTANDING.


I know that for beginner students, &nbsp;such problems are difficult and perplex them.

The only way to crawl into this area is to get it from an expert and read and re-read it as many times,
as it is needed for full understanding.  &nbsp;&nbsp;There is &nbsp;NO &nbsp;OTHER &nbsp;WAY.



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If you want to see other similar and different solved problems, &nbsp;look into the lesson

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Selected-problems-from-the-archive-on-calculating-trig-functions-of-angles.lesson>Advanced problems on calculating trigonometric functions of angles</A>

in this site.