General solutions question (1 Viewer)

[iStudent]

This has been on my mind for a while, but I never bothered to ask an explanation

For the general solution:



Using the yr11 formula, we should get



But some textbooks have: (cambridge)

(iirc)

Why does this happen?
edit: and how did they get this? is there a special technique or is it only that there is this special property
And are there also other trig functions with special cases? (e.g. other cos solutions?)
I know this is actually MX1 but this is seen mainly in MX2 (complex/polys)
Someone please explain, because this has been on my mind ever since I did complex numbers

 

[Shadowdude]

Well... the answers are equivalent. You can even do it graphically, and... yeah.

You can even say a solution is:




Which is the same thing again.

 

[iStudent]

Yea but how did they get that
that's what I want to know
(sorry if I didn't make myself clear)

 

[braintic]

Learning three different formulae for general solution is absolutely pointless.

Why does it happen that way in your particular question?

Write out the answers: pi/2, 3pi/2, 5pi/2, 7pi/2, ...

You can't see the pattern?

pi/2, pi + pi/2, 2pi + pi/2, 3pi + pi/2, ...

 

[iStudent]

Learning three different formulae for general solution is absolutely pointless.

Why does it happen that way in your particular question?

Write out the answers: pi/2, 3pi/2, 5pi/2, 7pi/2, ...

You can't see the pattern?

pi/2, pi + pi/2, 2pi + pi/2, 3pi + pi/2, ...

I know it works, but I don't know how they cut the general solution down to something without signs lol
That's the bit I don't get

 

[Shadowdude]

I know it works, but I don't know how they cut the general solution down to something without signs lol
That's the bit I don't get

It's the same thing, as I said.


You can express the set in a variety of ways, but they're all the same.

 

[iStudent]

I know they're the same thing (and I understand they're equivalent) but like
Why is it only that's so special to have many general solutions?
Are there any others? (e.g. I can't seem to find any for sin theta = 0)
And if there are, how do you know how to find them?

Do you understand what I'm trying to get across? :s

The one for is really useful because the textbook version is very complicated (and it takes much longer to find unique solutions if you have a sign in it as well. So I really want to know if it exists for others and how to find it

 

[Shadowdude]

I know they're the same thing (and I understand they're equivalent) but like
Why is it only that's so special to have many general solutions?
Are there any others? (e.g. I can't seem to find any for sin theta = 0)
And if there are, how do you know how to find them?

Do you understand what I'm trying to get across? :s

The one for is really useful because the textbook version is very complicated (and it takes much longer to find unique solutions if you have a sign in it as well. So I really want to know if it exists for others and how to find it

I think you'll find sin and tan have similar expressions.

 

[Squar3root]

because:

in one case 2n and equal n in another case.

example:

let n = 1 for

theta = 2*Pi*1 - Pi/2 = 3*Pi/2

but now let n = 4 for

theta = 4*Pi - Pi/2 = 3*Pi/2

which are the same thing

to be more correct, you should say theta over the Z field

 

[braintic]

I know they're the same thing (and I understand they're equivalent) but like
Why is it only that's so special to have many general solutions?
Are there any others? (e.g. I can't seem to find any for sin theta = 0)
And if there are, how do you know how to find them?

Do you understand what I'm trying to get across? :s

The one for is really useful because the textbook version is very complicated (and it takes much longer to find unique solutions if you have a sign in it as well. So I really want to know if it exists for others and how to find it

Because 2pi - p/2 is the same as pi + pi/2.
This is not true for other angles.

 

[iStudent]

I kind of see how..
So there's no 1 step method of getting from the first gen solution to the second?
You have to recognise patterns and form your own, right?
(before, I thought that there was a special technique to deriving the second gen solution from the first.. =/)
Well in that case, I won't worry about it so much since cos theta = 0 is probably the only special one that is important anyways.

Thanks for the help