The value of the determinant | sinθ cosθ sin 2 θ sin(θ+(2 π)/(3)) cos(θ+(2 π)/(3)

The value of the determinant | sinθ cosθ sin 2 θ ...

The value of the determinant $\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$ (A) 0 (B) $\sin \theta$ (C) $\cos \theta$ (D) independent of $\theta$

The value of the determinant $\left|\begin{array}{ccc}\sin \theta & \cos \theta & \sin 2 \theta \\ \sin \left(\theta+\frac{2 \pi}{3}\right) & \cos \left(\theta+\frac{2 \pi}{3}\right) & \sin \left(2 \theta+\frac{4 \pi}{3}\right) \\ \sin \left(\theta-\frac{2 \pi}{3}\right) & \cos \left(\theta-\frac{2 \pi}{3}\right) & \sin \left(2 \theta-\frac{4 \pi}{3}\right)\end{array}\right|$
(A) 0
(B) $\sin \theta$
(C) $\cos \theta$
(D) independent of $\theta$

Key Concepts

Determinants

Determinants are scalar values that can be computed from a square matrix and provide important information about the matrix, such as whether it is invertible and the volume scaling factor of the linear transformation represented by the matrix. They are widely used in solving systems of linear equations, finding eigenvalues, and understanding matrix properties like linear dependence among rows or columns.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. These identities, such as the sine and cosine addition formulas, double angle formulas, and periodic properties, allow for the simplification and transformation of expressions involving trigonometric functions, which is essential when working with matrices containing such functions.

Symmetry and Periodicity

Symmetry and periodicity in trigonometric functions refer to the inherent repeating and reflective properties of sine and cosine functions over specific intervals. These properties can often be exploited to simplify complex expressions or reveal invariant characteristics, such as the independence of a determinant from a variable, by showing that the matrix structure repeats or has symmetrical elements.

Angle Addition Formulas

Angle addition formulas, such as sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b ? sin a sin b, are used to break down trigonometric functions of sums or differences of angles into products of sine and cosine functions of the individual angles. This is particularly useful in problems where the matrix entries involve phase shifts or rotations.

Transcript

00:01 Problem 10 we want to find the value of the following determinant sign theta sign of theta plus 2 by divided by 3 sign of theta minus 2 by 2 by 3 then we have here cosine theta cosine theta plus 2 by 3 cosine theta minus 2 by 2 by 3 the 3 the 3 the 3 the 3 is sine 2 theta we have sign 2 theta plus 4 by divided by 3 and finally we have sine 2 theta minus 4 by divided by 3 you want to evaluate this determinant we can notice here we have only sine theta cosine theta sine 2 theta and the angles are the same 2 theta 2 theta and here we have only theta theta theta theta theta and we can reach only sine theta cosine theta sine 2 theta by taking the third row and added to the second row now we can do that and what is the result the result is sine plus sine cosine plus cosine sine and from the trigonometry properties we can get sign multiplied by cosine out of the summations let's do it we have sine theta cosine theta sine 2 theta taking the third row adding it to the second one sign plus sign gives us 2 multiplied by sine the summation divided by 2 we sum but inside the sign here and here then divide by 2 then it's sign 2 theta divided by 2 then it's theta multiplied by cosine the difference divided by 2 then it the difference gives 4 by 3 sorry the difference is 4 by 3 divided by 2 gives 2 by 2 by 2 by 2 we do the same the summation of cosine is 2 multiplied by cosine is 2 multiplied by cosine the sum divided by 2 then it's cosine theta 2 theta divided by 2 gives theta to apply it by cosine the difference divided by 2 is 2 by 5 by 3 finally we have 2 multiplied by sign 4 theta divided by 2 gives 2 2 2 2 2 by 2 2 2 by 2 2 2 2 by 2 2 2 2 2 2 3 by 2 and the 3 3 is as it is let's simplify the second row cosine 2x2 divided by 3 is minus half multiplied by 2 gives minus then equals sine theta cosine theta sine 2 theta then we have minus half multiplied by 2 gives minus sine seta here we have minus cosine theta here we have minus sine 2 theta and the third row as it is cosine theta minus 2 by divided by 3 and sine 2 theta minus 4 by divided by 3...

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