# Simplify of cosinus and sinus terms

• Jan 17th 2009, 11:22 AM
Herbststurm
Simplify of cosinus and sinus terms
Hi,

I have a calculation where I have to simplify some trogonometric terms. I did it up to here und and to here I am sure that it es correct but now I stumble. How to go on?

Target of the calculation is: $\displaystyle r^2 \sin(\Theta)$

This is the point I stumble:

$\displaystyle r^2 [ \cos(\Theta)^2 \sin(\Theta) \cos(\varphi)^2 + \sin(\Theta)^3 \sin(\varphi)^2$ $\displaystyle + \cos(\Theta)^2 \sin(\varphi)^2 \sin(\Theta) + \sin(\Theta)^3 \cos(\varphi)^2 ]$

Whats next?

I guess I have to use that sin square plus cos square is one, but I guess here it would be wrong.

thanks
greetings
• Jan 17th 2009, 11:28 AM
Mush
Quote:

Originally Posted by Herbststurm
Hi,

I have a calculation where I have to simplify some trogonometric terms. I did it up to here und and to here I am sure that it es correct but now I stumble. How to go on?

Target of the calculation is: $\displaystyle r^2 \sin(\Theta)$

This is the point I stumble:

$\displaystyle r^2 [ \cos(\Theta)^2 \sin(\Theta) \cos(\varphi)^2 + \sin(\Theta)^3 \sin(\varphi)^2$ $\displaystyle + \cos(\Theta)^2 \sin(\varphi)^2 \sin(\Theta) + \sin(\Theta)^3 \cos(\varphi)^2 ]$

Whats next?

I guess I have to use that sin square plus cos square is one, but I guess here it would be wrong.

thanks
greetings

Take out a factor of $\displaystyle \sin{\Theta}$

$\displaystyle r^2\sin{\Theta} [ \cos(\Theta)^2 \cos(\varphi)^2 + \sin(\Theta)^2 \sin(\varphi)^2$ $\displaystyle + \cos(\Theta)^2 \sin(\varphi)^2 + \sin(\Theta)^2 \cos(\varphi)^2 ]$

$\displaystyle r^2\sin{\Theta} [ (\cos(\Theta) \cos(\varphi))^2 + (\sin(\Theta) \sin(\varphi))^2$ $\displaystyle + (\cos(\Theta) \sin(\varphi))^2 + (\sin(\Theta) \cos(\varphi))^2 ]$

Can you now simplify the rest of the terms in the bracket to 1?

You can use the following to change the terms left over:

$\displaystyle \cos(\Theta \pm \varphi) = \cos(\Theta)\cos(\varphi) \mp \sin(\Theta)\sin(\varphi)$

$\displaystyle \sin(\Theta \pm \varphi) = \sin(\Theta)\cos(\varphi) \pm \cos(\Theta)\sin(\varphi)$

To use these to find identities for the terms you have, you have to add or subtract these equations from each other, depending on what you want.
• Jan 17th 2009, 11:34 AM
red_dog
$\displaystyle \cos^2\theta\sin\theta\cos^2\phi+\sin^3\theta\sin^ 2\phi+\cos^2\theta\sin^2\phi\sin\theta+\sin^3\thet a\cos^2\phi=$

$\displaystyle =\cos^2\theta\sin\theta(\cos^2\phi+\sin^2\phi)+\si n^3\theta(\sin^2\phi+\cos^2\phi)=$

$\displaystyle =\cos^2\theta\sin\theta+\sin^3\theta=\sin\theta(\c os^2\theta+\sin^2\theta)=\sin\theta$
• Jan 17th 2009, 12:24 PM
Herbststurm