Determinant of Matrix Involving trig Functions

In summary, the task is to find the determinant of the matrix {{cos 25°, sin 65°}, {sin 120°, cos 390°}} and the suggested approach is to use trigonometric identities to simplify the expression. The conversation also mentions the possibility of using rotation matrices, but no applicable properties were found. The participants then discuss using the identity cos(a + b) = (cos a)(cos b) - (sin a)(sin b) and suggest converting sin(65) and sin(120) into cosine functions. The conversation ends with a reminder to keep expanding the expression.
  • #1
V0ODO0CH1LD
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Homework Statement



Find the determinant of the matrix {{cos 25°, sin° 65}, {sin 120°, cos 390°}} (sorry, can't latex). {cos 25°, sin° 65} is first row and {sin 120°, cos 390°} is the second one.

Homework Equations



cos(a + b) = (cos a)(cos b) - (sin a) (sin b)

The Attempt at a Solution



I know you can just plug the values in a calculator, but apparently you can solve it by using some trig identities. I also though about using some property of rotation matrices but couldn't find any that fit the problem. Anyway, this is as far as I got:

cos (a + b) = (cos 25°)(cos 390°) - (sin 65°) (sin 120°)
 
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  • #2
Convert sin(65) into cos(something), rewrite sin(120) as sin(90+30), you can still do one more thing, cos(360+x)=cos(x). Can you start from here?
 
  • #3
Yeah, thanks! I was so caught up thinking on how to simplify the expression I forgot I could just keep on expanding it!
 

Related to Determinant of Matrix Involving trig Functions

1. What is the purpose of finding the determinant of a matrix involving trig functions?

The determinant of a matrix involving trig functions is used to determine whether the matrix is invertible or singular. It is also used to solve systems of equations and to calculate areas and volumes in geometry.

2. How is the determinant of a matrix involving trig functions calculated?

The determinant is calculated by using the properties of determinants and the cofactor expansion formula. The matrix is expanded along a row or column, and each element is multiplied by its corresponding cofactor. The sum of these products gives the determinant.

3. Can the determinant of a matrix involving trig functions be negative?

Yes, the determinant can be negative. The sign of the determinant is determined by the number of row or column operations needed to get the matrix in upper triangular form. If an odd number of operations are needed, the determinant will be negative.

4. Is there a specific method for solving determinants of matrices involving trig functions?

Yes, there are specific methods for solving these determinants, such as using trig identities or converting the trig functions to complex exponentials. However, the method used may vary depending on the complexity of the matrix and the desired level of accuracy.

5. Can a matrix involving trig functions have a zero determinant?

Yes, a matrix involving trig functions can have a determinant of zero. This means that the matrix is singular and has no inverse. It also indicates that the rows or columns are linearly dependent, and the system of equations represented by the matrix has infinitely many solutions.

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