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Question: If \(\frac{\pi}{2} < \alpha < \pi ⥂ ,\ \pi < \beta < \frac{3\pi}{2};\) \(\sin\alpha = \frac{15}{17}\...

If π2<α<π⥂, π<β<3π2;\frac{\pi}{2} < \alpha < \pi ⥂ ,\ \pi < \beta < \frac{3\pi}{2}; sin⁡α=1517\sin\alpha = \frac{15}{17} and tan⁡β=125\tan\beta = \frac{12}{5}, then the value of sin⁡(β−α)\sin(\beta - \alpha) is

A

– 171/221

B

– 21/221

C

21/221

D

171/221

Answer

171/221

Explanation

Solution

Given, sin⁡α=1517,tan⁡β=125\sin\alpha = \frac{15}{17},\tan\beta = \frac{12}{5}

⇒cos⁡α=817,sin⁡β=1213\Rightarrow \cos\alpha = \frac{8}{17},\sin\beta = \frac{12}{13}and cos⁡β=−513\cos\beta = - \frac{5}{13}

⇒ π<β<3π2\pi < \beta < \frac{3\pi}{2}, ∴cos⁡β=−513\therefore\cos\beta = - \frac{5}{13}

sin⁡(β−α)=sin⁡βcos⁡α−cos⁡βsin⁡α\sin(\beta - \alpha) = \sin\beta\cos\alpha - \cos\beta\sin\alpha = 171221\frac{171}{221}.