Angle Addition Formula for three angles?

[PFStudent]

Homework Statement

How do I find,

$$sin\left(\alpha - \beta + \gamma\right) = ?$$

Homework Equations

$$sin\left(\alpha\pm\beta\right) = sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta \pm cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta$$

and

$$cos\left(\alpha\pm\beta\right) = cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta \mp sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta$$

The Attempt at a Solution

$$sin\left(\alpha - \beta + \gamma\right) = ?$$

I know how to do it for four distinct angles,

$$sin\left(\alpha + \beta + \gamma + \psi\right) = ?$$

Where, let

$$\alpha + \beta = \theta$$

$$\gamma + \psi = \phi$$

And then expand, using the earlier identity I mentioned,

$$sin\left(\theta + \phi\right) = sin\theta{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\phi + cos\theta{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\phi$$

$$sin\left(\theta + \phi\right) = sin(\alpha + \beta){\textcolor[rgb]{1.00,1.00,1.00}{.}}cos(\gamma + \psi) + cos(\alpha + \beta){\textcolor[rgb]{1.00,1.00,1.00}{.}}sin(\gamma + \psi)$$

$$sin\left(\theta + \phi\right) = [sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta + cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi - sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi] + [cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta - sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi + cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi]$$

$$sin\left((\alpha + \beta) + (\gamma + \psi)\right) = [sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta + cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi - sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi] + [cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta - sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi + cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi]$$

$$sin\left(\alpha + \beta + \gamma + \psi\right) = [sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta + cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi - sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi] + [cos\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\beta - sin\alpha{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\beta]{\textcolor[rgb]{1.00,1.00,1.00}{.}}[sin\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}cos\psi + cos\gamma{\textcolor[rgb]{1.00,1.00,1.00}{.}}sin\psi]$$

However, for three angles, is where I am stumped.

Any help is appreciated.

Thanks,

-PFStudent

 

[Doc Al]

If you can do four, why can't you do three?

Let

$$\alpha + \beta = \theta$$

$$\gamma = \gamma$$

 

[PFStudent]

Hey,

Thanks for the quicky reply Doc Al, I hesitated to do that because was not sure if the folowing was true,

$$sin\left(\alpha + \beta + \gamma\right) = sin\left((\alpha + \beta) + \gamma\right) = sin\left(\alpha + (\beta + \gamma)\right)$$

The reason I ask is if any quantity (in the parentheses) can be let equal theta and expanded will they all be equal?

That is where I was unsure. That if you took each scenario I mentioned,

$$sin\left((\alpha + \beta) + \gamma\right)$$

$$sin\left(\alpha + (\beta + \gamma)\right)$$

And let the quantity in parentheses equal theta and applied the angle addition formula, would they still all be equal?

Or does it matter which pair of angles you let equal theta (i.e. does the answer change if you pick two different pairs)?

Thanks,

-PFStudent

 

[Dick]

Unfortunately, you also did the four angle one wrong. First expand $$sin\left(\theta + \phi\right)$$ and then put the definitions of theta and phi in and keep expanding. Each term should have trig functions of four angles in it. You expanded $$sin\left(\theta \right)+ sin\left(\phi\right)$$.

 

[Doc Al]

That is where I was unsure. That if you took each scenario I mentioned,

$$sin\left((\alpha + \beta) + \gamma\right)$$

$$sin\left(\alpha + (\beta + \gamma)\right)$$

And let the quantity in parentheses equal theta and applied the angle addition formula, would they still all be equal?

They better be! (That's the associative property of addition.)

Or does it matter which pair of angles you let equal theta (i.e. does the answer change if you pick two different pairs)?

Try it and see!

Unfortunately, you also did the four angle one wrong.

Thanks for checking, Dick. (I obviously didn't.)

 

[PFStudent]

Hey,

Thanks for the help guys, I edited my original post to reflect the correct expansion for angle addition of four angles.

Thanks,

-PFStudent