# Thread: Trigonometric identities in matrix

1. ## Trigonometric identities in matrix

Please can anyone shed some light on this?

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.

2. Originally Posted by stawbelly
Please can anyone shed some light on this?

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

3. 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!

4. 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.