Even Function Parameter Given where is an even function, find .

Trigonometry Solution – Problem 12: find

Even Function Parameter

Given $f(x) = \sin\left(\frac{x}{3} + \varphi\right)$ where $0 \leq \varphi < 2\pi$ is an even function, find $\varphi$ .

For $f(x)$ to be even, we need $f(-x) = f(x)$ for all $x$ :

$\sin\left(-\frac{x}{3} + \varphi\right) = \sin\left(\frac{x}{3} + \varphi\right)$

This requires $\cos\left(\frac{\varphi}{3}\right) = 0$ , so:

$\frac{\varphi}{3} = k\pi + \frac{\pi}{2}, \quad k \in \mathbb{Z}$

$\varphi = 3k\pi + \frac{3\pi}{2}, \quad k \in \mathbb{Z}$

Since $0 \leq \varphi < 2\pi$ , we have $\varphi = \frac{3\pi}{2}$ .

$\frac{3\pi}{2}$

Translate the question into conditions (2 pts): domain/range/period/symmetry/monotonicity constraints are written correctly.
Key transformation or rewrite (2 pts): rewrite the function into a standard form (e.g. amplitude/phase/period) or reduce using identities.
Correct interval/parameter reasoning (2 pts): derive the correct inequalities or argument interval.
Final answer (1 pt): state the final set / interval / period clearly.

Listing properties ("periodic", "even") without applying them to the given function.