general solution of trig equation

http://www.piscoau.net/000.jpg

Every things is alright now, but the reference answer is n*pi+[(-1)^n-1]*pi/4, which is different from mine, could anyone tell me how to combine that 2 solutions I found together?

http://www.piscoau.net/111.jpg

Again, a little problem, I found that the second general solution is strange because it make sec 5x + csc 2x undefined in some cases of n.

Re: general solution of trig equation

Honestly, I don't understand your question. Is this what you have done? Can you be more specific about what's your solutions etc ... ?

Re: general solution of trig equation

sorry for that...I have posted the wrong picture, now corrected

for the first question, I just want to know how to find the solution which is n*pi+[(-1)^n-1]*pi/4, where n is an integer. Is there something wrong with my process?

for the second question, I think I have solved the equation correctly, but the solution (2n*pi/3 - pi/2) can't satisfy the equation when n=3..etc

Re: general solution of trig equation

Quote:

Originally Posted by

**piscoau** http://www.piscoau.net/000.jpg
Every things is alright now, but the reference answer is n*pi+[(-1)^n-1]*pi/4, which is different from mine, could anyone tell me how to combine that 2 solutions I found together?

Hi piscoau,

For the first problem your answer is correct and the given answer is an equivalent. For,

$\displaystyle x=n\pi+\left[(-1)^n-1\right]\frac{\pi}{4}\mbox{ where }n\in Z$

When n is even, $\displaystyle n=2k~;~k\in Z$

$\displaystyle x=2k\pi+\left[(-1)^{2k}-1\right]\frac{\pi}{4}=2k\pi\mbox{ where }k\in Z$

When n is odd, $\displaystyle n=2k+1~;~k\in Z$

$\displaystyle x=(2k+1)\pi+\left[(-1)^{2k+1}-1\right]\frac{\pi}{4}=(2k+1)\pi-\frac{\pi}{2}=2k\pi+\frac{\pi}{2}\mbox{ where }k\in Z$

Hence, $\displaystyle x=2k\pi\mbox{ or }x=2k\pi+\frac{\pi}{2}\mbox{ where }k\in Z$

Quote:

Originally Posted by

**piscoau** for the second question, I think I have solved the equation correctly, but the solution (2n*pi/3 - pi/2) can't satisfy the equation when n=3..etc

Yes you have solved it correctly and it satisfies the equation for n=3.