If sin (theta + alpha ) = a and sin (theta + | vedclass

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

Question

(B) Given $\sin (	heta + \alpha ) = a$ and $\sin (	heta + \beta ) = b$.Let $x = 	heta + \alpha$ and $y = 	heta + \beta$. Then $\alpha - \beta = x

Vedclass · Accepted Answer

$1 - 2a^2 - 2b^2$

Vedclass · Answer

(B) Given $\sin (	heta + \alpha ) = a$ and $\sin (	heta + \beta ) = b$.Let $x = 	heta + \alpha$ and $y = 	heta + \beta$. Then $\alpha - \beta = x - y$.We have $\sin x = a$ and $\sin y = b$. Thus $\cos x = \pm \sqrt{1-a^2}$ and $\cos y = \pm \sqrt{1-b^2}$.$\cos (\alpha - \beta ) = \cos (x - y) = \cos x \cos y + \sin x \sin y = \pm \sqrt{1-a^2}\sqrt{1-b^2} + ab$.Let $C = \cos (\alpha - \beta )$. Then $C - ab = \pm \sqrt{1-a^2}\sqrt{1-b^2}$.Squaring both sides: $(C - ab)^2 = (1-a^2)(1-b^2) = 1 - a^2 - b^2 + a^2b^2$.$C^2 - 2abC + a^2b^2 = 1 - a^2 - b^2 + a^2b^2$.$C^2 - 2abC = 1 - a^2 - b^2$.We need to evaluate $\cos 2(\alpha - \beta ) - 4ab\cos (\alpha - \beta ) = 2\cos^2 (\alpha - \beta ) - 1 - 4ab\cos (\alpha - \beta )$.$= 2(C^2 - 2abC) - 1$.Substituting $C^2 - 2abC = 1 - a^2 - b^2$:$= 2(1 - a^2 - b^2) - 1 = 2 - 2a^2 - 2b^2 - 1 = 1 - 2a^2 - 2b^2$.

If $\sin (\theta + \alpha ) = a$ and $\sin (\theta + \beta ) = b,$ then $\cos 2(\alpha - \beta ) - 4ab\cos (\alpha - \beta )$ is equal to

Similar Questions

If $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 46^{\circ} \sin 47^{\circ}}+\ldots$ up to $45$ terms $=\frac{1}{\sin x^{\circ}}$,then $\sin \left(\frac{\pi}{2} x\right)=$

If $\cos x + \cos y = \frac{2}{3}$ and $\sin x - \sin y = \frac{3}{4}$,then find the value of $\sin(x - y) + \cos(x - y)$.

$\frac{\cos 13^{\circ}-\sin 13^{\circ}}{\cos 13^{\circ}+\sin 13^{\circ}}+\frac{1}{\cot 148^{\circ}}$ is equal to

The value of $\tan 20^{\circ} \tan 80^{\circ} \cot 50^{\circ} = $

$\sin ^4 \frac{\pi}{8}+\sin ^4 \frac{3 \pi}{8}+\sin ^4 \frac{5 \pi}{8}+\sin ^4 \frac{7 \pi}{8} = ?$

Vedclass Test Series

Exam Paper Generator

Online Exam Module