If a=cosθ+i sinθ, b=cos2 θ-i sin2 θ, c=cos3 θ+i sin3 θand if | a b c b c

If a=cosθ+i sinθ, b=cos2 θ-i sin2 θ, c=cos3 θ+i sin3 θand if...

If $a=\cos \theta+i \sin \theta, b=\cos 2 \theta-i \sin 2 \theta, c=\cos 3$ $\theta+i \sin 3 \theta$ and if $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$ then $\theta$ is equal to (A) $\overline{n \pi}$ (B) $2 n \pi$ (C) $(2 n+1) \frac{\pi}{2}$ (D) None of these

If $a=\cos \theta+i \sin \theta, b=\cos 2 \theta-i \sin 2 \theta, c=\cos 3$
$\theta+i \sin 3 \theta$ and if $\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$ then $\theta$ is equal to
(A) $\overline{n \pi}$
(B) $2 n \pi$
(C) $(2 n+1) \frac{\pi}{2}$
(D) None of these

Key Concepts

Euler's Formula

Euler's Formula provides the fundamental connection between complex exponentials and trigonometric functions by expressing e^(i?) as cos? + i sin?. This relationship is crucial for representing rotations in the complex plane and simplifying expressions involving trigonometric functions through exponential identities.

Matrix Determinants

The determinant of a square matrix is a scalar that encapsulates many properties of the matrix, including whether the matrix is invertible. A zero determinant indicates that the rows (or columns) of the matrix are linearly dependent, a fact that is often used to derive conditions or relationships among the entries of the matrix.

Circulant Matrices

Circulant matrices are a special class of matrices where each row is a cyclic shift of the previous one. These matrices have a structured form that allows their determinants and eigenvalues to be computed explicitly using the properties of complex exponentials and roots of unity, making them a powerful tool in solving problems with repetitive or symmetric patterns.

Transcript

00:01 I have given here the value of a, b, and c, and we're given determinant is equal to zero.

00:06 We're going to find the value of theta.

00:08 So just work on the determinant here.

00:14 Determinant we just apply here on this, let's say r1, tends to r1 plus r2, r2.

00:21 R3 we get a determinant is coming up to be a plus b plus b, a plus b, a plus b, a plus b, a plus b.

00:30 So p is b, t, b, t a, b, a b, b, that equals 0.

00:37 We can factor out from r1, a plus b plus b, get 1 -1, b, c -a, c -a, c -a -b.

00:50 That equals 0.

00:54 Next what we can do, we apply c2, tends to c2 negative t1, and c3 tends to c3 negative b1.

01:04 So, from here we get, we'll be a plus b plus b, time we'll have 1, b, b cc, column 1.

01:12 Column is 0, c negative b, a negative st, 0, a negative b, b negative c.

01:21 That equals 0.

01:24 So from here we get a plus b plus t expanding along r1, we get c negative b, b negative b negative b, negative b, a negative b, that equals 0.

01:45 If you now work on this, let's work on this.

01:49 As simplified, we get this relation, negative half a plus b plus c times 2a square plus 2b plus 2b plus 2b plus 2b plus 2c2c, that equals 0.

02:04 Next from here we get negative half a plus b plus c plus c times we have a negative b whole square plus c whole square plus c negative a whole square plus c negative a whole square that equals 0.

02:18 So from here, we just get here the one factor, a plus b plus c equal to 0.

02:26 So that means we'll have cost theta plus cost 3 theta plus 2 theta plus iota will have sine theta, sine 3 theta, negative sine 2 theta.

02:47 That equal 0.

02:47 So the real part is 0 we get cos theta plus cos 3 theta plus cos 2 theta that equals 0.

02:58 Applying here, cos a plus plus plus p over 2, cos a negative b over 2 and cos 2 theta equal to 0...

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