<PRE><B>Question 1117560</B>
&lt;br&gt;
By far the easiest way to solve the problem is to graph y=sin(x) and y = -0.493 on a graphing calculator over the specified interval and find where the graphs intersect.&lt;br&gt;
But most likely that solution method will not be acceptable on an exam....&lt;br&gt;
On the other hand, it would be helpful, if allowed, to find the solution that way so you can check the answer you get using your knowledge of the behavior of the sine function.&lt;br&gt;
So here is a graph of the sine function from -pi to +2pi, along with a graph of y=-0.493:&lt;br&gt;
{{{graph(400,400,-pi,2pi,-1,1,sin(x),-0.493)}}}&lt;br&gt;
On the specified interval, there is clearly a solution at about x = 3.5; in fact, my graphing calculator gives the value as x = 3.6571272.  And it looks as if there might be a solution at about x=6; but  it turns out that that intersection is just outside the specified interval.&lt;br&gt;
So your job is to find the single solution at about x = 3.5.&lt;br&gt;
Clearly you need to use your calculator to find an angle x for which sin(x)=-0.493.  Most (all?) calculators will show {{{sin^-1(-0.493) = -.5155345573}}} to several decimal places.&lt;br&gt;
You can see that on the above graph -- the first intersection point to the left of x=0.  Your task is to use that value to find the solution at about x = 3.5, using what you know about the sine function.&lt;br&gt;
There are many ways you could do that.  Perhaps another tutor will also respond to your question and use a different method than I&#39;m going to use.  That could be helpful to you; perhaps you would find a different method than mine to be more to your liking.&lt;br&gt;
So here is the reasoning I would use to find the answer.&lt;br&gt;
(1) The value of x at about 3.5 where sin(x) = -0.493 is exactly half a period (pi) past the point where sin(x) = +0.493.&lt;br&gt;
(2) The value of x where sin(x) = +0.493 is the opposite of the value where sin(x) = -0.493.&lt;br&gt;
So I will find my answer by taking the angle my calculator gives me, multiplying it by -1 to get the opposite of that angle, and then adding pi:&lt;br&gt;
{{{(-1)(-.5155345573)+pi = 3.657127211}}}&lt;br&gt;
Hooray!  That agrees with the answer I got by graphing.</PRE>