# Simplify of cosinus and sinus terms

• Jan 17th 2009, 12:22 PM
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: $r^2 \sin(\Theta)$

This is the point I stumble:

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

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, 12:28 PM
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: $r^2 \sin(\Theta)$

This is the point I stumble:

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

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 $\sin{\Theta}$

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

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

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:

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

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

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, 12:34 PM
red_dog
$\cos^2\theta\sin\theta\cos^2\phi+\sin^3\theta\sin^ 2\phi+\cos^2\theta\sin^2\phi\sin\theta+\sin^3\thet a\cos^2\phi=$

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

$=\cos^2\theta\sin\theta+\sin^3\theta=\sin\theta(\c os^2\theta+\sin^2\theta)=\sin\theta$
• Jan 17th 2009, 01:24 PM
Herbststurm