To find the reference number $t'$ for $t = \frac{54\pi}{6}$, follow these steps:

Step 1: Simplify the Given Angle

$t = \frac{54\pi}{6} = 9\pi$

Step 2: Find the Coterminal Angle in $[0, 2\pi]$

Since $9\pi$ is greater than $2\pi$, we reduce it by subtracting multiples of $2\pi$:

$9\pi - 4(2\pi) = 9\pi - 8\pi = \pi$

Thus, the coterminal angle of $9\pi$ in the standard interval $[0, 2\pi]$ is $\pi$.

Step 3: Find the Reference Number

• The reference number is the distance between the given angle and the closest multiple of $\pi$.
• Since $\pi$ lies on the x-axis (180° or $\pi$), the reference number is:

$t' = |t - \pi| = \pi - \pi = 0$

Final Answer:

$t' = 0$

Would you like further details or explanations?

Related Questions:

1. How do you find reference numbers for angles in different quadrants?
2. What is the significance of reference numbers in trigonometry?
3. How do you determine coterminal angles?
4. Why is the reference number always positive?
5. What happens if the given angle is negative?

Tip:

A reference number is always the smallest positive acute angle made with the x-axis.