# Write sin(x-(3pi)/4) in terms of sinx?

Apr 11, 2017

$\sin \left(x - \frac{3 \pi}{4}\right) = - \frac{1}{\sqrt{2}} \left(\sin x + \sqrt{1 - {\sin}^{2} x}\right)$

#### Explanation:

As $\sin \left(A - B\right) = \sin A \cos B - \cos A \sin B$

Hence $\sin \left(x - \frac{3 \pi}{4}\right) = \sin x \cos \left(\frac{3 \pi}{4}\right) - \cos x \sin \left(\frac{3 \pi}{4}\right)$

= $\sin x \cos \left(\pi - \frac{\pi}{4}\right) - \cos x \sin \left(\pi - \frac{\pi}{4}\right)$

= $- \sin x \cos \left(\frac{\pi}{4}\right) - \cos x \sin \left(\frac{\pi}{4}\right)$

= $- \sin x \times \frac{1}{\sqrt{2}} - \cos x \times \frac{1}{\sqrt{2}}$

= $- \frac{1}{\sqrt{2}} \left(\sin x + \cos x\right)$

= $- \frac{1}{\sqrt{2}} \left(\sin x + \sqrt{1 - {\sin}^{2} x}\right)$