To find the reference angle of $\frac{16\pi}{29}$, follow these steps:

Step 1: Determine the quadrant where the angle lies.

• A reference angle is the acute angle between the terminal side of the given angle and the x-axis.
• First, simplify $\frac{16\pi}{29}$ to determine how many radians it is relative to $2\pi$ (a full rotation).

$2\pi = \frac{58\pi}{29}.$

Thus: $\frac{16\pi}{29} < \frac{58\pi}{29}.$

Since $\frac{16\pi}{29}$ is less than $2\pi$, the angle lies in the first revolution. Now, divide $\frac{16\pi}{29}$ by $\pi$ to approximate its position in terms of half-revolutions ($\pi$):

$\frac{16\pi}{29} \div \pi = \frac{16}{29} \approx 0.55.$

This means $\frac{16\pi}{29}$ is slightly more than half a radian, and it lies in Quadrant I.

Step 2: Reference angle in Quadrant I.

• In Quadrant I, the reference angle is the angle itself: $\text{Reference Angle} = \frac{16\pi}{29}.$

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Final Answer:

The reference angle is: $\frac{16\pi}{29}.$

Would you like further clarifications on reference angles or how this calculation was done?

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Related Questions:

1. How do reference angles differ in Quadrants II, III, and IV?
2. What is the relationship between reference angles and trigonometric functions?
3. How would you convert $\frac{16\pi}{29}$ into degrees to check its approximate size?
4. What happens to the reference angle if the original angle exceeds $2\pi$?
5. How can we verify this computation using a calculator or software?

Tip:

To ensure accuracy, always determine the quadrant first, as it dictates how reference angles are calculated.