Why is there only one general solution instead of two?
you are solving for when sinx = 0.....x = n*pi 'cause that's when it equals zero.....and that covers ALL the solutions, not just one.....to convince urself, sub in when n =2....that will equal zero....n =3 etc
you are solving for when sinx = 0.....x = n*pi 'cause that's when it equals zero.....and that covers ALL the solutions, not just one.....to convince urself, sub in when n =2....that will equal zero....n =3 etc
The thing is, you're not going wrong. I think Lord of the Flies explained it quite clearly, so you might want to re-read his if you're still down here.
If you have trouble making the logical step of combining two solution sets, I think it would be easier to look at the period. For example, the period of sin2x is pi/2, so solutions should be pi/2 apart, and that's what your resulting "cycle length" should end up as. I'm not sure it works though...
Generally:
Unparseable latex formula:
\sin\theta=Q\Rightarrow\theta= $Arcsin$Q+2\pi k, \pi-$Arcsin$Q+2\pi k
.
While cosine is much nicer:
Unparseable latex formula:
\cos\phi=Q\Rightarrow\phi=\pm $Arccos$Q+2\pi k
This can all be derived from the unit circle. Alternatively, use this Wikipedia article as a resource.
Thanks for clearing it up but is there any sort of method as such that I can use to condense the "2" general solutions?
Or do I have to instinctively see it?
There is no method because you cannot always condense the two solutions.
For instance:
Spoiler
Also, it isn't the end of the world if you don't see the simplification. Your solution is still correct if you give both solutions separately. So in your case you would have written:
x={nπ,2π+nπ} which is fine. Condensing is just prettier and perhaps easier to visualise.
Thanks for clearing it up but is there any sort of method as such that I can use to condense the "2" general solutions?
Or do I have to instinctively see it?
I am going to add something:
If you end up needing to condense, (most of the time) it is that you didn't notice a simpler solution to start with. Take your equation, by picturing the circle you should be able to see that sin equals 0 with periodicity pi, not only 2pi. Essentially: