The question asks to find the reference angle for $\frac{4\pi}{15}$.

Solution:

The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis. Reference angles are always between $0$ and $\frac{\pi}{2}$ (or $0^\circ$ and $90^\circ$).

Steps:

1. Determine the quadrant: The angle $\frac{4\pi}{15}$ is in radians. To determine the quadrant, compare $\frac{4\pi}{15}$ with standard multiples of $\pi$:

   • $\pi \approx 3.14$, so:
     • $\frac{\pi}{2} = \frac{15\pi}{30} = \frac{7.5\pi}{15}$.
     • $\pi = \frac{15\pi}{15}$.

   Since $\frac{4\pi}{15}$ is less than $\frac{\pi}{2}$, the angle is in the first quadrant.

2. Reference angle in the first quadrant: In the first quadrant, the angle itself is the reference angle. Hence, the reference angle for $\frac{4\pi}{15}$ is: $\theta_{\text{ref}} = \frac{4\pi}{15}.$

Final Answer:

The reference angle is $\boxed{\frac{4\pi}{15}}$.

Would you like further explanation or a detailed breakdown of related concepts?

Expanding Questions:

1. What is the general formula for finding reference angles in each quadrant?
2. How do you convert between radians and degrees for this problem?
3. What is the geometric interpretation of reference angles?
4. What happens if the given angle exceeds $2\pi$ or is negative?
5. Can you apply reference angles to solve trigonometric functions like sine or cosine?

Tip:

Always remember that reference angles are non-negative and fall between $0$ and $\frac{\pi}{2}$ radians (or $0^\circ$ and $90^\circ$).