Question

The value of $\left|\begin{array}{ccc}\cos (\theta +\alpha )& \sin (\theta +\alpha )& 1\\ \cos (\theta +\beta )& \sin (\theta +\beta )& 1\\ \cos (\theta +v)& \sin (\theta +v)& 1\end{array}\right|$ is

• A. dependent on $\theta$
• B. independent of $\theta$
• C. always $\theta$
• D. cannot be determined

Answer 1

Answer: (B)

independent of $\theta$

Let a triangle ABC on unit circle have polar coordinates as $(r,{\theta }_1),(r,{\theta }_2),(r,{\theta }_3)$
where ${\theta }_1=\theta +\alpha$, ${\theta }_2=\theta +\beta$ ,${\theta }_3=\theta +\gamma$
So, the coordinates of A, B, C in cartesian form are $\left(\cos \right(\theta +\alpha ),\sin (\theta +\alpha \left)\right),\left(\cos \right(\theta +\beta ),\sin (\theta +\beta \left)\right),\left(\cos \right(\theta +\gamma ),\sin (\theta +\gamma \left)\right)$
Area of triangle ABC $=\frac{1}{2}\left|\begin{array}{ccc}cos(\theta +\alpha )& sin(\theta +\alpha )& 1\\ cos(\theta +\beta )& sin(\theta +\beta )& 1\\ cos(\theta +v)& sin(\theta +v)& 1\end{array}\right|$
Now, since the value of $\theta$ is changing, that means we are rotating the triangle.
But the area of triangle remains same while rotation.
Hence, the area is independent of $\theta$
So, the given determinant is independent of $\theta$