# Express sin(x-(3pi)/4) as a function of sinx?

Jan 31, 2018

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

#### Explanation:

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

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

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

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

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

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