# Trigonometric identities in matrix

• Sep 25th 2006, 11:34 AM
stawbelly
Trigonometric identities in matrix
Please can anyone shed some light on this? :confused: :(

I would like to use trigonometric identities to simplify the 2x2 matrix in the attached word document.

I have also included an example calculation from a paper but I don't understand how the simplified matrix was determined from the trig identities. Could anyone explain this to me?

Any help would be much appreciated.
• Sep 25th 2006, 11:44 AM
topsquark
Quote:

Originally Posted by stawbelly
Please can anyone shed some light on this? :confused: :(

I would like to use trigonometric identities to simplify the 2x2 matrix in the attached word document.

I have also included an example calculation from a paper but I don't understand how the simplified matrix was determined from the trig identities. Could anyone explain this to me?

Any help would be much appreciated.

Consider the 1,1 component of the example:
sin(x)sin(z)cos(y) - cos(x)sin(y)

We are told sin(z) = -1, so this is equal to
-sin(x)cos(y) - cos(x)sin(y) = -[sin(x)cos(y) + cos(x)sin(y)]

which is, according to identity 1 at the top of the page
-sin(x + y)

The rest of them are pretty much the same.

For example in the question, look at the 1,2 component:
-cos(x)sin(z)cos(y) + sin(x)sin(y)

Again, sin(z) = -1 so this is equal to:
cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

-Dan
• Sep 25th 2006, 12:30 PM
stawbelly
Many thanks for your clear explanation.

I have solved for the other components of the problem:

1,1 component

sin(x)sin(z)cos(y) + cos(x)sin(y)
= -sin(x)cos(y) + cos(x)sin(y) = -sin(x-y) according to identity 3

1,2 component (as shown in your post)

-cos(x)sin(z)cos(y) + sin(x)sin(y)
= cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

2,1 component

-sin(x)sin(z)sin(y) + cos(x)cos(y)
= sin(x)sin(y) + cos(x)cos(y) = cos(x-y) according to identity 4

2,2 component

cos(x)sin(z)sin(y) + sin(x)cos(y)
= -cos(x)sin(y) + sin(x)cos(y) = sin(x-y) according to identity 3

I was a bit unsure about component 1,1 but if any of them are wrong, please let me know :)

thanks!
• Sep 25th 2006, 12:40 PM
ThePerfectHacker
Quote:

Originally Posted by stawbelly
Many thanks for your clear explanation.

I have solved for the other components of the problem:

1,1 component

sin(x)sin(z)cos(y) + cos(x)sin(y)
= -sin(x)cos(y) + cos(x)sin(y) = -sin(x-y) according to identity 3

1,2 component (as shown in your post)

-cos(x)sin(z)cos(y) + sin(x)sin(y)
= cos(x)cos(y) + sin(x)sin(y) = cos(x - y) according to identity 4.

2,1 component

-sin(x)sin(z)sin(y) + cos(x)cos(y)
= sin(x)sin(y) + cos(x)cos(y) = cos(x-y) according to identity 4

2,2 component

cos(x)sin(z)sin(y) + sin(x)cos(y)
= -cos(x)sin(y) + sin(x)cos(y) = sin(x-y) according to identity 3

I was a bit unsure about component 1,1 but if any of them are wrong, please let me know :)

thanks!

They all look good to me.