Question 578708
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Find the angle(s) that would make each statement true.
theta equals arc csc (sqrt of 2)
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        The solution in the post by @Theo,  giving two different possible measures  45  degrees or  135  degrees 

        for the angle  {{{theta}}}   is incorrect.


        I came to bring a correct solution.



<pre>
{{{theta}}} = {{{arccsc(sqrt(2))}}}  means, by the definition


    {{{csc(theta)}}} = {{{sqrt(2)}}}  and  {{{-pi/2}}} <= {{{theta}}} <= {{{pi/2}}}.


For this standard definition, see, for example, this Wikipedia article

https://en.wikipedia.org/wiki/Inverse_trigonometric_functions



In turn, {{{csc(theta)}}}  is cosecant {{{theta}}}

    {{{csc(theta)}}} = {{{1/sin(theta)}}}.



So, the problem wants you find angle  {{{theta}}}  such that

    {{{1/sin(theta)}}} = {{{sqrt(2)}}}  and  {{{-pi/2}}} <= {{{theta}}} <= {{{pi/2}}}.



In other words, you want to find angle  {{{theta}}}  such that

    {{{sin(theta)}}} = {{{1/sqrt(2)}}} = {{{sqrt(2)/2}}}  and  {{{-pi/2}}} <= {{{theta}}} <= {{{pi/2}}}.



There is only one such angle, and this angle is the standard  {{{highlight(highlight(Table))}}}  of angles, 
which students learn in Trigonometry course.  Its measure is 

    {{{theta}}} = {{{pi/4}}}  in radians,  or  {{{theta}}} = 45 degrees.



<U>ANSWER</U>.  The angle  {{{theta}}}  satisfying the problem's condition is  {{{pi/4}}}  radians, or  45 degrees.
         Such an angle is  {{{highlight(highlight(UNIQUE))}}}.
</pre>

Solved.


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Perhaps, &nbsp;tens, &nbsp;and hundreds, &nbsp;and thousands students failed to answer this simple question on exam.


To answer correctly, &nbsp;it is extremely important to know firmly the definitions .