How to Find the Minimum Angle Value Involving Two Variables and Constraints?

[A13235378]

Homework Statement

Find the minimum angle value $\theta$ such that:

$\theta$= 2 $\theta_1$ - 4 $\theta_2$ + 180

$\theta_1$ and $\theta_2$ < 90 , $\theta_1$ > $\theta_2$

$\sin\theta_1=n \sin\theta_2$

Relevant Equations

n/a

I tried to do it by derivative but there are two variables, so I don't know how to proceed. Does anyone know how I can solve it?

Remembering that you don't need to find the value of $\theta$. I just need to find a relationship between $\theta_1$ and $\theta_2$

 

[Office_Shredder]

I'm confused. The first line says to find a value of $\theta$. Then at the end you say you don't need to figure out what $\theta$ is. What exactly do you need to solve for?

 

[A13235378]

I'm confused. The first line says to find a value of $\theta$. Then at the end you say you don't need to figure out what $\theta$ is. What exactly do you need to solve for?

I found an answer here in my book that says so that $\theta$ it is minimal

$sin\theta_1 = \sqrt{\frac{4-n^2}{3}}$

So how do i get to this

 

[Mark44]

Homework Statement:: Find the minimum angle value $\theta$ such that:

$\theta$= 2 $\theta_1$ - 4 $\theta_2$ + 180

$\theta_1$ and $\theta_2$ < 90 , $\theta_1$ > $\theta_2$

$\sin\theta_1=n \sin\theta_2$
Relevant Equations:: n/a

I tried to do it by derivative but there are two variables, so I don't know how to proceed. Does anyone know how I can solve it?

Remembering that you don't need to find the value of $\theta$. I just need to find a relationship between $\theta_1$ and $\theta_2$

I suspect that the actual wording of the problem is something like "Find the minimum value of $2\theta_2 - 4\theta_2 + 180$" subject to the constraints given.
I haven't worked the problem, but I would start with this:
$\sin(2\theta_2 - 4\theta_2 + 180) = -\sin(2\theta_2 - 4\theta_2) = \dots$
From there I would use the identity for the sine of a difference of angles; i.e., $\sin(A - B) = \sin(A)\cos(B) - \sin(B)\cos(A)$.
One of the constraints is $\sin(\theta_1) = n\sin(\theta_2)$, so that could be used to simplify things in the expression we started with.
I haven't worked this through, but what I've described is how I would approach the problem.

 

[ehild]

I suspect that the actual wording of the problem is something like "Find the minimum value of $2\theta_2 - 4\theta_2 + 180$" subject to the constraints given.
I haven't worked the problem, but I would start with this:
$\sin(2\theta_2 - 4\theta_2 + 180) = -\sin(2\theta_2 - 4\theta_2) = \dots$

I tried it, but found the method of Lagrange multipliers simpler.

 

[ehild]

Homework Statement:: Find the minimum angle value $\theta$ such that:

$\theta$= 2 $\theta_1$ - 4 $\theta_2$ + 180

$\theta_1$ and $\theta_2$ < 90 , $\theta_1$ > $\theta_2$

$\sin\theta_1=n \sin\theta_2$
Relevant Equations:: n/a

I tried to do it by derivative but there are two variables, so I don't know how to proceed. Does anyone know how I can solve it?

You have rwo minimize a two-variable function θ(θ1,θ2). What is the condition that a two-variable function has a local minimum or maximum? How to handle a comstraint? Have you learned the method of Lagrange mulriplier?