Trigonometry – Problem 13: find

Graph Translation and Symmetry

Translate $y = \sin x + \cos x$ left by $m$ units ($0 < m < \pi$). If the resulting graph is symmetric about the y-axis, find $m$.

First, rewrite the function:

$$y = \sin x + \cos x = \sqrt{2}\sin\left(x + \frac{\pi}{4}\right)$$

After translating left by $m$ units:

$$y = \sqrt{2}\sin\left(x + m + \frac{\pi}{4}\right)$$

For y-axis symmetry (even function):

$$m + \frac{\pi}{4} = k\pi + \frac{\pi}{2}, \quad k \in \mathbb{Z}$$

$$m = k\pi + \frac{\pi}{4}, \quad k \in \mathbb{Z}$$

Since $0 < m < \pi$, when $k = 0$: $m = \frac{3\pi}{4}$.

$\frac{3\pi}{4}$