# Minimum value of an angle

• A13235378

#### 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##

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?

• jim mcnamara
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

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.
##\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.

• jim mcnamara
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.
##\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.

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?