Simplify Equation - Can You Help?

[hermano]

Hi,

I'm trying to simplify the following equation, but I'm not sure it can be! Can anybody help me please?

$$\frac{sin(\alpha + \delta \alpha)} {sin(\beta + \delta \beta - \alpha - \delta \alpha)} - \frac{sin(\alpha)} {sin(\beta - \alpha)}$$

 

[berkeman]

Hi,

I'm trying to simplify the following equation, but I'm not sure it can be! Can anybody help me please?

$$\frac{sin(\alpha + \delta \alpha)} {sin(\beta + \delta \beta - \alpha - \delta \alpha)} - \frac{sin(\alpha)} {sin(\beta - \alpha)}$$

That's not an equation (no equal sign), it is an expression. What is the full equation?

What is the context of the question? Is it for schoolwork, or for a physical problem?

 

[hermano]

That's not an equation (no equal sign), it is an expression. What is the full equation?

What is the context of the question? Is it for schoolwork, or for a physical problem?

Oke, the full equation is
$$\epsilon = \frac{sin(\alpha + \delta \alpha)} {sin(\beta + \delta \beta - \alpha - \delta \alpha)} - \frac{sin(\alpha)} {sin(\beta - \alpha)}$$

It is for an error analyses of a physical problem!

Thanks

 

[I like Serena]

Welcome to PF, hermano!

Let's define $f(x) = {\sin(\alpha x) \over sin(\gamma x)}$.
Do you know what the definition of f'(1) is?

 

[CellCoree]

Sure, I'd be happy to help you simplify this equation. Let's start by simplifying the first fraction:

\frac{sin(\alpha + \delta \alpha)} {sin(\beta + \delta \beta - \alpha - \delta \alpha)} = \frac{sin(\alpha)cos(\delta \alpha) + cos(\alpha)sin(\delta \alpha)} {sin(\beta)cos(\delta \beta) - cos(\beta)sin(\delta \beta) - sin(\alpha)cos(\delta \alpha) - cos(\alpha)sin(\delta \alpha)}

Next, we can use the fact that sin(x) = cos(\pi/2 - x) to simplify the second fraction:

\frac{sin(\alpha)} {sin(\beta - \alpha)} = \frac{cos(\pi/2 - \alpha)} {cos(\pi/2 - (\beta - \alpha))} = \frac{cos(\pi/2 - \alpha)} {cos(\alpha - \pi/2 + \beta)} = \frac{cos(\pi/2 - \alpha)} {sin(\alpha + \beta)}

Now, we can substitute these simplified fractions back into the original equation and combine like terms to get the final simplified form:

\frac{sin(\alpha + \delta \alpha)} {sin(\beta + \delta \beta - \alpha - \delta \alpha)} - \frac{sin(\alpha)} {sin(\beta - \alpha)} = \frac{cos(\alpha)sin(\delta \alpha)} {sin(\beta)cos(\delta \beta) - cos(\beta)sin(\delta \beta) - sin(\alpha)cos(\delta \alpha) - cos(\alpha)sin(\delta \alpha)} - \frac{cos(\pi/2 - \alpha)} {sin(\alpha + \beta)}

I hope this helps simplify the equation for you. Let me know if you have any further questions or need clarification.