# 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.

• I've never used TEX before, and couldn't find a way to make the degree symbol - any help would be appreciated. I'm also unsure in how many steps of my solving to show, more can of course be presented if needed.
– Zar
Dec 8, 2013 at 16:09
• You can make a degree symbol with ^\circ Dec 8, 2013 at 18:56
• @Arthur Thanks! Updated OP.
– Zar
Dec 8, 2013 at 18:59

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.

• My original calculations divided 360 by 6, forgot that when I wrote the answer. Are you sure? Because this is an 'official' exam question from a respected univeristy. I'm not good enough at math to dare say that they're wrong. :-)
– Zar
Dec 8, 2013 at 17:20
• @Zar Sapere aude! For the missing $22.6$ convince the world that it's a solution. For the solution beyond $90$: well, they are greater than $90$, thus not in the scope of our task. There are exactly four solutions to the given problem. Dec 8, 2013 at 18:04
• Hehe good point. Many thanks for your time, I'll talk to my professor tomorrow and return with the response. (Cannot tag you using @.., by the way)
– Zar
Dec 8, 2013 at 19:10
• My answers were the correct, and he had no idea how those answers came upon that sheet. Many thanks for your help, once again!
– Zar
Dec 9, 2013 at 20:05
• Congrats! Please feel free to upvote. Dec 10, 2013 at 6:27

Don't you think that it would be better if
$10sin(6x) = 7 \iff x = \begin{cases} 7.4+n*360 / 6 \ \\ 22.6 + n*360 / 6 \end{cases}$

• Oh, obviously. In my original calculation, I of course divided 360 by 6. Did the math in my head while writing the question. Updated OP, the question still remains, though.
– Zar
Dec 8, 2013 at 17:17