Question

Evaluate : sin 7pi/12 . cos pi/4 - cos 7pi/12 . sin pi/4

Answer 1

Sin(A-B) = SinA*CosB - CosA*SinB
Sin($\frac{7 \pi }{12}$ - $\frac{ \pi }{4}$) = Sin($\frac{7 \pi }{12}$ )*Cos($\frac{\pi }{4}$ ) -  Cos($\frac{7 \pi }{12}$ )*Sin($\frac{\pi }{4}$ )
Sin($\frac{4 \pi }{12}$) = Sin($\frac{ \pi }{3}$ = $\frac{ \sqrt{3} }{2}$

Answer 2

Answer:

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\frac{\sqrt{3}}{2}$

Step-by-step explanation:

To evaluate : $\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}$

Solution :

We know that,

$\sin(A-B)=\sin A\cdot \cos B-\cos A\cdot\cos B$

On comparing with the given expression,

$A=\frac{7\pi}{12}$ and $B=\frac{\pi}{4}$

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\sin(\frac{7\pi}{12}-\frac{\pi}{4})$

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\sin(\frac{7\pi-3\pi}{12})$

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\sin(\frac{4\pi}{12})$

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\sin(\frac{\pi}{3})$

We know, $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$

$\sin\frac{7\pi}{12}\cdot\cos\frac{\pi}{4}-\cos\frac{7\pi}{12}\cdot\sin\frac{\pi}{4}=\frac{\sqrt{3}}{2}$