# Math Help - General solution help

1. ## General solution help

How do I answer questions like find all general solutions to :cos(x) = sin(x)?
I know I need to get 2 different equations like x = 2npi +/- ... for cos and x = npi + (-1)^n ... but how do I get it so that cos(x) = ... and sin(x) = ... so I can apply the formula to the equation

2. Hello,
Originally Posted by guest12345
How do I answer questions like find all general solutions to :cos(x) = sin(x)?
I know I need to get 2 different equations like x = 2npi +/- ... for cos and x = npi + (-1)^n ... but how do I get it so that cos(x) = ... and sin(x) = ... so I can apply the formula to the equation
If $\cos x=0$, $\sin x$ equals 1 or -1 so the equality $\cos x=\sin x$ can't hold and $\cos x$ must be different from 0. Thus we can safely divide both sides by $\cos x$ and it gives us a new equation: $1=
\frac{\sin x}{\cos x}\Longleftrightarrow 1=\tan x$
. Can you solve this ?

3. yea but that's not what I was meaning, I was meaning if there's an equation with sin(4x) = cos(5x) (as posted in my new thread) or something of that nature then how do i solve it not for the equation in the post above, was just meaning that sort of equation in general, sorry if I was unclear.

4. Originally Posted by guest12345
yea but that's not what I was meaning, I was meaning if there's an equation with sin(4x) = cos(5x) (as posted in my new thread) or something of that nature then how do i solve it not for the equation in the post above, was just meaning that sort of equation in general, sorry if I was unclear.
Then you can simply use $\cos x = \sin \left(x+\frac{\pi}{2}\right)
$
:

$
\sin(4x)=\cos(5x) \Longleftrightarrow \sin(4x)=\sin\left(5x+\frac{\pi}{2} \right)\Longleftrightarrow x\equiv\ldots$

5. Originally Posted by guest12345
yea but that's not what I was meaning, I was meaning if there's an equation with sin(4x) = cos(5x) (as posted in my new thread) or something of that nature then how do i solve it not for the equation in the post above, was just meaning that sort of equation in general, sorry if I was unclear.
This question has been answered as a consequence of solving the problem posted at http://www.mathhelpforum.com/math-he...-solution.html