The value of θ, lying between θ=0 and θ=(π)/(2) and satisfying the equation | 1+cos ^2θ si

step 1 Note that in the given problem, the given equation can be rewritten as determinant of 1 plus cos

The value of θ, lying between θ=0 and θ=(π)/(2) and...

The value of $\theta$, lying between $\theta=0$ and $\theta=\frac{\pi}{2}$ and satisfying the equation $\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0$, is (A) $\frac{11 \pi}{24}$ (B) $\frac{7 \pi}{24}$ (C) $\frac{5 \pi}{24}$ (D) none of these

The value of $\theta$, lying between $\theta=0$ and $\theta=\frac{\pi}{2}$ and satisfying the equation
$\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 4 \sin 4 \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+4 \sin 4 \theta\end{array}\right|=0$, is
(A) $\frac{11 \pi}{24}$
(B) $\frac{7 \pi}{24}$
(C) $\frac{5 \pi}{24}$
(D) none of these

Key Concepts

Matrix Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix and contains important information about the matrix. It is used to determine if the matrix is singular or non-singular, and its evaluation often involves techniques like row operations. In problems involving trigonometric entries, properties of determinants and strategic manipulations can simplify the calculation significantly.

Trigonometric Identities

Trigonometric identities are equalities involving trigonometric functions that hold for all values of the involved variables. They are essential in simplifying expressions with sine, cosine, and other trigonometric functions, particularly when these functions are part of more complex algebraic or matrix expressions. Recognizing and applying these identities can reduce the complexity of equations that otherwise might be difficult to solve directly.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the angles that satisfy the given trigonometric conditions while taking into account the periodic nature of trigonometric functions. Techniques include using identities to simplify the equation, isolating the trigonometric function, and then determining the solutions within the specified range. This concept is crucial for finding specific angle values that satisfy conditions derived from more complex arrangements like determinants.

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