Using Lagrange multipliers to find minimum value

Question

Use Lagrange multipliers to find the minimum value of

$$ T = \frac {a}{v \cos \alpha} + \frac {b}{v\cos\beta} $$

subject to the constraint

$$ L = a\tan \alpha + b\tan\beta $$

where $a, b, v$ and $L$ are all constants.

I have no idea how to do this.

Answer 1

Hint: Refer to this: http://en.wikipedia.org/wiki/Lagrange_multiplier

Then let $\alpha$, $\beta$ be the variables.

Answer 2

Smells like Snell's.

Hint: Let $g(\alpha, \beta) = a \tan\alpha + b \tan \beta - L$. You need to set up the following system of equations:

$$L = a \tan \alpha + b \tan \beta$$ $$\frac{\partial T}{\partial \alpha} = \lambda \frac{\partial g}{\partial \alpha}$$ $$ \frac{\partial T}{\partial \beta} = \lambda \frac{\partial g}{\partial \beta}$$

This is a system of $3$ equations with $3$ unknowns and your job is to try and find for what values of $\alpha, \beta$, and $\lambda$ this system is satisfied. However, when using Lagrange multipliers, you might want to think of $\lambda$ as a tool to get to the values of $\alpha$ and $\beta$ that make this system of equations consistent, since at the end of the day, you don't want the values of $\lambda$, but rather just the values of $\alpha$ and $\beta$. When you find the points $(\alpha, \beta)$ (there could be more than $1$ point), you can evaluate your function $T$ to see what values they give.