help: trigonometric equation

physicsRookie

[tex]x_1(cos\alpha-1) + x_2sin\alpha = 0 [/tex]
[tex]x_1sin\alpha + x_2(-cos\alpha-1) = 0 [/tex]
How to solve this equation? Can anyone help me?

cepheid

It's a system of equations: 2 equations in 2 unknowns. That means you can solve it. Just solve for one unknown in terms of the other using the first equation, and then subsitute that into the second.

physicsRookie

Let me try...

Solve equation 1:
[tex]x_2 = \frac{-x_1(cos\alpha - 1)}{sin\alpha}[/tex]

Substitute it to the second:
[tex]x_1sin\alpha + \frac{-x_1(cos\alpha - 1)}{sin\alpha}(-cos\alpha-1) = 0[/tex]

[tex]x_1sin\alpha + \frac{x_1(cos^2\alpha - 1)}{sin\alpha} = 0[/tex]

[tex]2x_1sin\alpha = 0[/tex]

What is the solutions for [tex]2x_1sin\alpha = 0[/tex]?
Obviously one is [tex]x_1=0[/tex], but if [tex]sin\alpha = 0[/tex], then...

HallsofIvy

Well done. The point is, of course, that [tex]\alpha[/tex] is a number (not one of the variables) so these can be solved like any pair of equations for x₁ and x₂.
Notice, by the way, that if [tex]sin\alpha= 0[/tex], your first step, dividing by that, would be invalid. You have to look at this case separately.
If [tex]sin\alpha= 0[/tex], then [tex]cos\alpha[/tex] is either 1 or -1.

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= 1[/tex]?

What do your equations look like if [tex]sin\alpha= 0[/tex] and [tex]cos\alpha= -1[/tex]?

physicsRookie

HallsofIvy, thanks.
[tex]2x_1=0 and 0=0 => x_1=0, x_2 [/tex] could be any number
[tex] 0=0 and -2x_2=0 => x_2=0, x_1 [/tex] could be any number

I just try another solution.
Rewrite the equations:
[tex](x_1cos\alpha + x_2sin\alpha) - x_1 = 0 [/tex]
[tex](x_1sin\alpha - x_2cos\alpha) - x_2 = 0 [/tex]

Suppose [tex]x_1 = cos\frac{\alpha}{2}, x_2 = sin\frac{\alpha}{2}[/tex], then

[tex]cos\frac{\alpha}{2}cos\alpha + sin\frac{\alpha}{2}sin\alpha - cos\frac{\alpha}{2} = 0 [/tex]

[tex]cos\frac{\alpha}{2}sin\alpha - sin\frac{\alpha}{2}cos\alpha - sin\frac{\alpha}{2} = 0 [/tex]

It works!

I am wondering whether there is some general method to solve [tex]x_1, x_2[/tex] depending on [tex]\alpha[/tex] or not.