Mark the tick against the correct answer in the following: |(sinα, cosα, sin(α + δ)),(sinβ, cosβ, sin(β + δ))(sin γ, cosγ, sin(α + δ)) = ?

Question

Mark the tick against the correct answer in the following:

\(\begin{vmatrix} sin \alpha & cos \alpha & sin(\alpha + \delta) \\[0.3em] sin \beta & cos \beta & sin(\beta + \delta) \\[0.3em] sin \gamma &cos \gamma & sin(\gamma + \delta) \end{vmatrix}\) = ?

|(sinα, cosα, sin(α + δ)),(sinβ, cosβ, sin(β + δ))(sin γ, cosγ, sin(α + δ)) = ?

A. 0 

B. 1 

C. sin (α + δ) + sin (β + δ)+ sin (γ + δ) 

D. none of these

Answer 1

Correct answer A. 0

To find: value of \(\begin{vmatrix} sin \alpha & cos \alpha & sin(\alpha + \delta) \\[0.3em] sin \beta & cos \beta & sin(\beta + \delta) \\[0.3em] sin \gamma &cos \gamma & sin(\gamma + \delta) \end{vmatrix}\)

Formula Used: sin(A+B) = sinAcosB+cosAsinB

We have, \(\begin{vmatrix} sin \alpha & cos \alpha & sin(\alpha + \delta) \\[0.3em] sin \beta & cos \beta & sin(\beta + \delta) \\[0.3em] sin \gamma &cos \gamma & sin(\gamma + \delta) \end{vmatrix}\)

Applying C₁ → cos (δ) C₁

= 0 

When two columns are identical then the value of determinant is 0