1. ## Solving some equations

Hello.

Does anyone know how to solve these equations:

$\displaystyle X*cos(t*a) + Y*sin(t*a) = 0$

$\displaystyle X*cos(t*b) + Y*sin(t*b) = 0$

I want to find nontrivial solutions for X,Y and t. I only know that $\displaystyle a \not= 0 \not= b$. Furthermore $\displaystyle a,b \in \mathbb{R}$

Help would be much appreciated.

Rapha

2. Hello, Rapha!

Does anyone know how to solve these equations?

. . $\displaystyle \begin{array}{ccccc}x\cos(at) + y\sin(at) &=& 0 & [1] \\ x\cos(bt) + y\sin(bt) &=& 0 & [2] \end{array}$

I want to find nontrivial solutions for $\displaystyle x,y$ and $\displaystyle t.$
I only know that $\displaystyle a,b \neq 0$. .Furthermore: $\displaystyle a,b \in \mathbb{R}$

$\displaystyle \begin{array}{ccccccc} \text{Multiply [1] by }\sin(bt)\!: & x\cos(at)\sin(bt) + y\sin(at)\sin(bt) &=& 0 & [3] \\ \text{Multiply [2] by }\sin(at)\!: & x\sin(at)\cos(bt) + y\sin(at)\sin(bt) &=& 0 & [4] \end{array}$

Subtract [4] - [3]: .$\displaystyle x\sin(at)\cos(bt) - x\cos(at)\sin(bt) \;=\;0$

. . . . . . . . . . . . . $\displaystyle x\bigg[\sin(at)\cos(bt) - \cos(at)\sin(bt)\bigg] \;=\;0$

. . . . . . . . . . . . . . . . . . . . . . . . . .$\displaystyle x\sin(at - bt) \;=\;0$

We have: .$\displaystyle x \:=\:0 \quad \hdots \text{ trivial solution}$

. . And: .$\displaystyle \sin(a-b)t \;=\;0 \quad\Rightarrow\quad (a-b)t \;=\;\pi n \quad\Rightarrow\quad t \;=\;\frac{\pi n}{a-b}$

3. Hello Soroban.

That was really helpful, thank you very much!

Kind regards
Rapha