Evaluate: sin (beta + gamma - alpha ) + sin ( | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let the expression be $E = \sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \b

Vedclass · Accepted Answer

$4\sin \alpha \sin \beta \sin \gamma $

Vedclass · Answer

(B) Let the expression be $E = \sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma )$.Using the sum-to-product formula $\sin C + \sin D = 2 \sin \frac{C+D}{2} \cos \frac{C-D}{2}$:Group the first two terms: $\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) = 2 \sin \gamma \cos (\beta - \alpha )$.Group the last two terms: $\sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma ) = 2 \cos (\alpha + \beta ) \sin (-\gamma) = -2 \sin \gamma \cos (\alpha + \beta )$.Combining these,$E = 2 \sin \gamma [\cos (\beta - \alpha ) - \cos (\alpha + \beta )]$.Using the identity $\cos (A-B) - \cos (A+B) = 2 \sin A \sin B$:$E = 2 \sin \gamma [2 \sin \alpha \sin \beta] = 4 \sin \alpha \sin \beta \sin \gamma $.

Evaluate: $\sin (\beta + \gamma - \alpha ) + \sin (\gamma + \alpha - \beta ) + \sin (\alpha + \beta - \gamma ) - \sin (\alpha + \beta + \gamma )$

Similar Questions

Prove that $\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \cot x$.

If $\cos \theta = \frac{8}{17}$ and $\theta$ lies in the $1^{st}$ quadrant,then the value of $\cos (30^\circ + \theta) + \cos (45^\circ - \theta) + \cos (120^\circ - \theta)$ is

If $\tan \theta = \frac{\cos 25^{\circ} + \sin 25^{\circ}}{\cos 25^{\circ} - \sin 25^{\circ}}$ and $\theta$ is in the third quadrant,then $\theta =$ (in $^{\circ}$)

$\cos \alpha \sin (\beta-\gamma) + \cos \beta \sin (\gamma-\alpha) + \cos \gamma \sin (\alpha-\beta)$ is equal to

Let $\tan \alpha = \frac{a}{a+1}$ and $\tan \beta = \frac{1}{2a+1}$, then $\alpha + \beta$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module