Q:

Solve $10sin(6x) = 7$ in the interval $0 < x < 90$, solutions go beyond $90$?

I'm to find all solutions to the equation:
$10\sin(6x) = 7$ in the interval $0^\circ <x<90^\circ $.
What I've got so far:
$10sin(6x) = 7 
\iff
x = \begin{cases} 
7.4 + n * 60^\circ  \
\\
22.6 + n * 60^\circ  
\end{cases}$
where n $\in Z$
Here, only $0$ and $1$ are possible values for $n$ to get $x$ to remain within the given interval, which gives the solutions:
$x_{1,2,3,4} = \begin{cases} 
7.4^\circ , 67.4^\circ 
\\
22.6^\circ , 82.6^\circ 
\end{cases}$
However, the answer sheet lists the following solutions:
$7.4^\circ , 67.4^\circ , 82.6^\circ , 127.4^\circ , 142.6^\circ $. Also note that $22.6^\circ $ is NOT listed as a solution here, and the solutions go way beyond $90^\circ $, which was the end of the interval. They also answer with 5 values, not identical to mine.
What am I missing? My solving method is obviously incorrect.

A:

Since $\arcsin(0.7)=44.427$ you know that $6x=44.427+360n$ or $6x=135.573+360n$, hence $x=7.405+60n$ or $x=22.595+60n$.  You'll make it from here.  The values of $x>90$ are clearly wrong answers in the scope of the question.

