Q.Find the value of $\theta$ satisfying $\begin{vmatrix} 1 & 1 & \sin 3\theta \\ -4 & 3 & \cos 2\theta \\ 7 & -7 & -2 \end{vmatrix} = 0$.

Step 1 — Expand the determinant. Expanding $\Delta = \begin{vmatrix} 1 & 1 & \sin 3\theta \\ -4 & 3 & \cos 2\theta \\ 7 & -7 & -2 \end{vmatrix}$ along the first row:

Step 2 — Set $\Delta=0$. Dividing by 7: $-2+2\cos 2\theta+\sin 3\theta=0$, i.e. $\sin 3\theta+2\cos 2\theta=2$.

Step 3 — Reduce to one function. Using $\cos 2\theta=1-2\sin^2\theta$ and $\sin 3\theta=3\sin\theta-4\sin^3\theta$:

$3\sin\theta-4\sin^3\theta+2-4\sin^2\theta=2 \;\Rightarrow\; \sin\theta(-4\sin^2\theta-4\sin\theta+3)=0.$ …

When a determinant equals zero, it means the rows (or columns) are linearly dependent. But here, we're not looking for a dependency relationship — we're solving for a specific variable $\theta$ that makes the determinant vanish. The direct approach is to expand the determinant and simplify the resulting trigonometric equation.

The determinant is $3 \times 3$, so expansion is straightforward. The key is to handle the trigonometric terms carefully and use identities to reduce everything to a single trigonometric function.

Step 1: Expand the determinant

Let

Expanding along the first row (or any row/column — choose what's simplest):

Step 2: Compute each $2 \times 2$ determinant

First minor:

Second minor (note the minus sign in front of the cofactor):

Third minor:

Step 3: Assemble the expansion

Simplify:

Factor 7:

Set $\Delta = 0$:

So:

Step 4: Use trigonometric identities to reduce to one function

We have $\sin 3\theta$ and $\cos 2\theta$. Use:

• $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$
• $\cos 2\theta = 1 - 2\sin^2\theta$

Substitute:

Simplify:

Cancel the 2 on both sides:

Factor $\sin\theta$:

$\sin\theta (3 - 4\sin^2\theta - 4\sin\theta) = 0$ …

Method: Solving for an Unknown Angle from a Determinant Equation

When a determinant containing trigonometric functions of an unknown angle is set equal to zero, the method is to expand the determinant into a plain trigonometric equation first, then solve that equation using standard multiple-angle identities.

Steps

Step 1: Expand the determinant using cofactor expansion

Pick the row or column with the simplest (constant) entries and expand along it. This converts the determinant condition into a linear combination of the trigonometric quantities appearing in the matrix (here, $\sin$ of a multiple angle and $\cos$ of another multiple angle), multiplied by fixed numeric coefficients from the $2\times2$ minors.

Step 2: Set the expanded expression equal to zero and simplify

Collect like terms and divide through by any common numeric factor to reach the cleanest possible trigonometric equation, something of the form $p\sin(m\theta)+q\cos(n\theta)=r$.

Step 3: Rewrite every term in terms of a single trig function …

Common Mistakes

Mistake 1: Dropping the alternating sign in cofactor expansion

Expanding a $3\times3$ determinant along a row, the middle term always carries a $-$ sign. Missing it here turns the correctly-expanded $-14+14\cos2\theta+7\sin3\theta$ into a wrong expression and every step after it is built on a false equation.

Mistake 2: Misremembering the triple-angle identity

Using $\sin3\theta=3\sin\theta-\sin^3\theta$ instead of the correct $\sin3\theta=3\sin\theta-4\sin^3\theta$ corrupts the cubic equation in $\sin\theta$ and leads to entirely wrong roots. …