To solve these problems, we'll find both the reference number and the terminal point on the unit circle for each value of $t$.

Understanding Reference Number and Terminal Point

1. Reference Number: It is the acute angle between the terminal side of the given angle $t$ and the x-axis. It helps in simplifying trigonometric calculations.
2. Terminal Point: It is the point on the unit circle corresponding to the angle $t$.

Part (i): $t = \frac{7\pi}{3}$

1. Convert $t$ to an equivalent angle between $0$ and $2\pi$: $\frac{7\pi}{3} = 2\pi + \frac{\pi}{3}$ So, the equivalent angle within $0$ to $2\pi$ is $\frac{\pi}{3}$.

2. Reference Number: Since $\frac{\pi}{3}$ is already in the first quadrant, the reference number is $\frac{\pi}{3}$.

3. Terminal Point: The terminal point for $\frac{\pi}{3}$ is: $\left( \cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right) \right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right).$

Part (ii): $t = -\frac{11\pi}{6}$

1. Convert $t$ to an equivalent positive angle: $-\frac{11\pi}{6} + 2\pi = -\frac{11\pi}{6} + \frac{12\pi}{6} = \frac{\pi}{6}.$

2. Reference Number: The reference number is $\frac{\pi}{6}$, as the angle is in the first quadrant.

3. Terminal Point: The terminal point for $\frac{\pi}{6}$ is: $\left( \cos\left(\frac{\pi}{6}\right), \sin\left(\frac{\pi}{6}\right) \right) = \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right).$

Part (iii): $t = \frac{17\pi}{4}$

1. Convert $t$ to an equivalent angle between $0$ and $2\pi$: $\frac{17\pi}{4} = 4\pi + \frac{\pi}{4} = 2\pi + 2\pi + \frac{\pi}{4}.$ So, the equivalent angle is $\frac{\pi}{4}$.

2. Reference Number: Since $\frac{\pi}{4}$ is in the first quadrant, the reference number is $\frac{\pi}{4}$.

3. Terminal Point: The terminal point for $\frac{\pi}{4}$ is: $\left( \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right) \right) = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right).$

Summary

1. $t = \frac{7\pi}{3}$:

   • Reference Number: $\frac{\pi}{3}$
   • Terminal Point: $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$

2. $t = -\frac{11\pi}{6}$:

   • Reference Number: $\frac{\pi}{6}$
   • Terminal Point: $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$

3. $t = \frac{17\pi}{4}$:

   • Reference Number: $\frac{\pi}{4}$
   • Terminal Point: $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$

Would you like more details or have any questions?

Related Questions:

1. How do you find the reference angle for negative values of $t$?
2. What is the relationship between reference angles and their corresponding trigonometric values?
3. How does adding $2\pi$ affect the terminal point?
4. Can a reference number ever be greater than $\pi/2$?
5. How do terminal points help in understanding the unit circle?

Tip:

Always convert angles to a standard range (usually $0$ to $2\pi$) to easily identify the reference angle and terminal point.