Question 1117560
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For this problem,  the algorithm of the solution is as follows.


<pre>
1.  You should understand that the solutions are in quadrants QIII and QIV.


2.  Calculate the given endpoints of the solution interval 


    {{{pi/6}}} = {{{3.14/6}}} = 0.523;   {{{7pi/4}}} = {{{(7*3.14)/4}}} = 5.495.


3.  Find  arcsin(0.493) = 0.516 radians (using your calculator).  It is in quadrant QI.


4.  You should understand that the solution to  sin(b) = - 0.493  is {{{pi}}} units (half the period of sine) ahead in QIII:  

        {{{b[1]}}} = arcsin(0.493) + {{{pi}}} = 0.516 + 3.14 = 3.656.


    It is still in the assigned limits.  Thus you just found the solution in QIII.



5.  Now you need to find (and to check) the solution in QIV.


    There are two ways to find it.


        a)  You can find arcsin(-0.493), using your calculator. You will get  

            {{{b[2]}}} = -0.516 radians.

            Add  {{{2pi}}} = 6.28  (one rotation about the unit circle) to it to get QIV.

            So, your new value for {{{b[2]}}} is -0.516 + 6.28 = 5.764.

            But it is greater than the limit 5.495 of the given domain, so this solution does not work.


          b)  Or, knowing the behavior of sine function,  you can find the distance from {{{b[1]}}}  to  {{{3pi/2}}}:  

              {{{3pi/2 - 3.656}}} = {{{(3*3.14)/2 - 3.656}}} = 1.054

              and then to add this value 1.054  to  {{{3pi/2}}}:  {{{(3*3.14)/2 + 1.054}}} = 5.764  ( ! the SAME value as you found above ! )

              It is your candidate for  {{{b[2]}}};  but it fails since it is greater than the upper boundary of  5.495.
</pre>

Your analysis is completed.  &nbsp;&nbsp;You just found the unique solution  {{{b[1]}}} = 3.656.


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So, your algorithm again:


<pre>
     - Find the boundaries of the domain.

     - Think in what quadrants the solution should be.

     - Evaluate the solution using your calculator.

     - Check if it satisfies the given constraints.
</pre>

Surely, &nbsp;you should know the behavior &nbsp;/&nbsp; (how the plots look like) &nbsp;for basic trigonometry functions.