Angle between two straight lines

[arpon]

Homework Statement

Find the angle between the straight lines:
$(x^2+y^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(x \tan{\theta} - y \sin{\alpha})^2$

Homework Equations

[Not applicable]

The Attempt at a Solution

Dividing by $x^2$,
$(1+(\frac{y}{x})^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - \frac{y}{x} \sin{\alpha})^2$
Let, $\frac{y}{x} =m$.
$(1+m^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - m \sin{\alpha})^2$
$(\sin^2{\theta} \cos^2{\alpha}) m^2 + (2 \tan{\theta} \sin{\alpha}) m + (\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta}-\tan^2{\theta}) = 0$
So the solutions of this equation indicate the slopes of the two straight lines. If the solutions are $m_1$ and $m_2$, then the angle between the two straight lines will be $\arctan{\frac{m_1-m_2}{1+m_1 m_2}}$;
I came up with a messy equation, as I tried to calculate this. But the answer is very simple, just $2\theta$. So, I think there is some clever technique to solve this problem.
Any suggestion will be appreciated

 

[Mark44]

Homework Statement

Find the angle between the straight lines:
$(x^2+y^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(x \tan{\theta} - y \sin{\alpha})^2$

Homework Equations

[Not applicable]

The Attempt at a Solution

Dividing by $x^2$,
$(1+(\frac{y}{x})^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - \frac{y}{x} \sin{\alpha})^2$
Let, $\frac{y}{x} =m$.
$(1+m^2)(\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta})=(\tan{\theta} - m \sin{\alpha})^2$
$(\sin^2{\theta} \cos^2{\alpha}) m^2 + (2 \tan{\theta} \sin{\alpha}) m + (\cos^2{\theta} \sin^2{\alpha} + \sin^2{\theta}-\tan^2{\theta}) = 0$
So the solutions of this equation indicate the slopes of the two straight lines. If the solutions are $m_1$ and $m_2$, then the angle between the two straight lines will be $\arctan{\frac{m_1-m_2}{1+m_1 m_2}}$;
I came up with a messy equation, as I tried to calculate this. But the answer is very simple, just $2\theta$. So, I think there is some clever technique to solve this problem.
Any suggestion will be appreciated

Instead of dividing by x^2, I think you might be better off by replacing x^2 + y^2 by r^2, and replacing x and y by r*cos(θ) and r*sin(θ), respectively. This would get the equation completely into polar form, after which it might be easier to simplify.

 

[arpon]

Instead of dividing by x^2, I think you might be better off by replacing x^2 + y^2 by r^2, and replacing x and y by r*cos(θ) and r*sin(θ), respectively. This would get the equation completely into polar form, after which it might be easier to simplify.

Thanks for your reply. This also gives me same kind of equation.

 

[Mark44]

Thanks for your reply. This also gives me same kind of equation.

Doing what I suggested, I get a somewhat simpler equation in polar form.

Is there more to this problem than you have posted? Is there an explanation of what $\theta$ and $\alpha$ represent?

Also, is the equation you wrote exactly the same as given in the problem?

 

[arpon]

Doing what I suggested, I get a somewhat simpler equation in polar form.

Is there more to this problem than you have posted? Is there an explanation of what $\theta$ and $\alpha$ represent?

Also, is the equation you wrote exactly the same as given in the problem?

No, there is no explanation about $\theta$ & $\alpha$. And the equation I wrote is exactly the same as in the problem.

 

[Mark44]

Can you post an image of the problem? Your solution and my solution both seem to come out very messy, and I'm wondering if there is something we're not seeing.