| 1+cos ^2θ sin ^2θ 2 √(3)tanθ cos ^2θ 1+sin ^2θ 2 √(3)tanθ cos ^2θ sin

| 1+cos ^2θ sin ^2θ 2 √(3)tanθ cos ^2θ 1+sin...

$\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 2 \sqrt{3} \tan \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 2 \sqrt{3} \tan \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+2 \sqrt{3} \tan \theta\end{array}\right|=0$, if $\theta$ is equal to (i) $\frac{\pi}{6}$ (ii) $\frac{5 \pi}{6}$ (iii) $\frac{7 \pi}{6}$ (iv) $\frac{11 \pi}{6}$ (a) (i) or (ii) (b) (ii) or (iii) (c) (ii) or (iv) (d) (i) or (iv)

$\left|\begin{array}{ccc}1+\cos ^{2} \theta & \sin ^{2} \theta & 2 \sqrt{3} \tan \theta \\ \cos ^{2} \theta & 1+\sin ^{2} \theta & 2 \sqrt{3} \tan \theta \\ \cos ^{2} \theta & \sin ^{2} \theta & 1+2 \sqrt{3} \tan \theta\end{array}\right|=0$, if $\theta$ is equal to
(i) $\frac{\pi}{6}$
(ii) $\frac{5 \pi}{6}$
(iii) $\frac{7 \pi}{6}$
(iv) $\frac{11 \pi}{6}$
(a) (i) or (ii)
(b) (ii) or (iii)
(c) (ii) or (iv)
(d) (i) or (iv)

Key Concepts

Determinants

A determinant is a scalar value that can be computed from the elements of a square matrix. It plays a key role in linear algebra, particularly in solving systems of linear equations, analyzing matrix invertibility, and studying eigenvalues. The ability to expand and simplify determinants, sometimes using cofactor expansion or by applying matrix properties, is essential when dealing with matrices that include functions or variable expressions.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in understanding relationships in right triangles and circular motion. Their periodic behavior, defined by specific angle values, allows them to model cyclic phenomena. Recognizing how these functions interact in equations, especially when squared terms like sin² and cos² appear, is crucial for simplifying and solving trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within the domain of the functions. Identities such as sin²? + cos²? = 1 are especially useful for simplifying expressions that contain multiple trigonometric functions. These identities help to reduce complex expressions to simpler forms, facilitating the solution of equations that may initially appear intricate due to the mixture of different functions.

Special Angle Values

Special angle values refer to specific angles, such as those commonly given in radians or degrees, where trigonometric functions yield exact and often simplifying results. Understanding these angles is integral in both solving trigonometric equations and verifying solutions, as they often reveal symmetries and periodic properties in the functions. This knowledge is useful for determining which angles satisfy equations derived from either purely algebraic or hybrid algebraic-trigonometric contexts.

Transcript

00:01 So in the given question what we have over here is we have a determinant which is given as 1 plus cos square theta, sine square theta, 2 root 3 times tan theta and we have in the second row cos square theta 1 plus sine square theta 2 root 3 tan theta and and in the third row we have cost square theta, sine square theta, and 1 plus 2 root 3 tan theta.

00:49 And we are told that this determinant has a value which is equal to 0.

00:55 And if that's the case, we are told to find the value of theta.

00:59 So what we can do over here is we can do some row operations in this matrix in this determinant in order to simplify it so let's change r1 let's change r1 to r1 minus r2 and let's change r2 to r2 minus r3 so if we do this transformations row transformations, row operations on this determinant what we would have is 1 minus 1 0, 0 1 minus 1 and cos square theta, sine square theta 1 plus 2 root 3 tant theta.

01:51 So this is what we would have as the determinant now.

01:56 Next what we can do is we already know that this determinant has a value which is equal to 0.

02:05 So let's add the columns c2 and c1 next.

02:11 So if you do the column operation that is c2 changes to c2 plus c1 then what we would have is we would have 1 minus 1 minus 1 plus 1 is 0 so 1 0 0 and 0 and 0 and 0 1 plus 0 is 0 itself 1 plus 0 is 1 plus 0 is 1 and we have cos square theta over here and cost square theta plus sine square theta over here we should know the identity that cost square theta plus sine square theta is equal to 1 so that we can substitute 1 over here and the rest is the same right c 2 changes to c 2 plus c1 so minus 1 plus 1 plus 1 1 plus 0 and cosquare theta plus sine square theta is what we have in the second column and now we can evaluate this determinant.

03:16 So this would be equal to 1 plus 2 root 3 tan theta minus minus 1 which is equal to 2...

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