# How do you find the x intercepts of y=sinpix+cospix?

Feb 20, 2018

$x = k - \frac{1}{4} \text{ }$ for any integer $k$

#### Explanation:

Note that:

$y = \sin \pi x + \cos \pi x$

$\textcolor{w h i t e}{y} = \sqrt{2} \left(\frac{\sqrt{2}}{2} \sin \pi x + \frac{\sqrt{2}}{2} \cos \pi x\right)$

$\textcolor{w h i t e}{y} = \sqrt{2} \left(\sin \left(\frac{\pi}{4}\right) \sin \pi x + \cos \left(\frac{\pi}{4}\right) \cos \pi x\right)$

$\textcolor{w h i t e}{y} = \sqrt{2} \sin \left(\frac{\pi}{4} + \pi x\right)$

Also note that:

$\sin \theta = 0 \text{ }$ if and only if $\theta = k \pi$ for some integer $k$

So we require:

$\frac{\pi}{4} + \pi x = k \pi$

Dividing both sides by $\pi$, this becomes:

$\frac{1}{4} + x = k$

Then subtracting $\frac{1}{4}$ from both sides we find:

$x = k - \frac{1}{4} \text{ }$ for any integer $k$