To find the reference angle for the angle $\frac{29\pi}{3}$, we first need to bring the angle within the standard range of $0$ to $2\pi$.

Step 1: Reduce the angle to within one full rotation

Since $2\pi$ represents a full rotation, we want to subtract multiples of $2\pi$ from $\frac{29\pi}{3}$ until it falls within the range $0 \leq \theta < 2\pi$.

Start by dividing $\frac{29\pi}{3}$ by $2\pi$ to see how many full rotations fit: $\frac{29\pi}{3} \div 2\pi = \frac{29}{6}$ This gives approximately $4.833$. This means that $\frac{29\pi}{3}$ has completed 4 full rotations, and we have a remainder of $0.833$ of a full rotation.

Step 2: Subtract full rotations

Now, subtract the integer part of the division (which is 4 full rotations, or $4 \times 2\pi = 8\pi$) from $\frac{29\pi}{3}$: $\frac{29\pi}{3} - 8\pi = \frac{29\pi}{3} - \frac{24\pi}{3} = \frac{5\pi}{3}$ So, the reduced angle is $\frac{5\pi}{3}$, which is between $0$ and $2\pi$.

Step 3: Find the reference angle

The reference angle is the smallest positive angle between the terminal side of the given angle and the x-axis. To find it for an angle in the fourth quadrant (like $\frac{5\pi}{3}$), we subtract it from $2\pi$:

$2\pi - \frac{5\pi}{3} = \frac{6\pi}{3} - \frac{5\pi}{3} = \frac{\pi}{3}$

Thus, the reference angle for $\frac{29\pi}{3}$ is $\frac{\pi}{3}$.

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Would you like further details or have any questions?

Here are some related questions for you to explore:

1. How do you find the reference angle for an angle in the second quadrant?
2. How can you convert an angle in radians to degrees?
3. What is the relationship between an angle and its coterminal angles?
4. How do you find the reference angle for $\frac{7\pi}{4}$?
5. How do periodic functions (like sine and cosine) behave for angles greater than $2\pi$?

Tip: When working with angles larger than $2\pi$, always reduce the angle to a value between $0$ and $2\pi$ first to make finding the reference angle easier.