Example: Find the terminal points on the unit circle determined by \( t=3 \pi, t=-\pi \), and \( t=-\frac{\pi}{2} \)

To find the terminal points on the unit circle determined by the angles \( t = 3\pi \), \( t = -\pi \), and \( t = -\frac{\pi}{2} \), we will follow these steps: 1. **Understanding the Unit Circle**: The unit circle is defined as the set of all points \((x, y)\) such that \(x^2 + y^2 = 1\). The angle \(t\) is measured in radians from the positive x-axis. 2. **Finding the Equivalent Angles**: Since angles on the unit circle are periodic with a period of \(2\pi\), we can find equivalent angles within the range \([0, 2\pi)\) by adding or subtracting \(2\pi\) as necessary. 3. **Calculating the Terminal Points**: - For each angle, we will calculate the coordinates \((x, y)\) using the formulas: \[ x = \cos(t) \] \[ y = \sin(t) \] Now, let's calculate the terminal points for each angle. ### Step 1: Calculate for \( t = 3\pi \) First, we find the equivalent angle: \[ 3\pi - 2\pi = \pi \] Now, we calculate the coordinates: \[ x = \cos(\pi) = -1 \] \[ y = \sin(\pi) = 0 \] Thus, the terminal point for \( t = 3\pi \) is \((-1, 0)\). ### Step 2: Calculate for \( t = -\pi \) Next, we find the equivalent angle: \[ -\pi + 2\pi = \pi \] Now, we calculate the coordinates: \[ x = \cos(\pi) = -1 \] \[ y = \sin(\pi) = 0 \] Thus, the terminal point for \( t = -\pi \) is also \((-1, 0)\). ### Step 3: Calculate for \( t = -\frac{\pi}{2} \) Now, we find the equivalent angle: \[ -\frac{\pi}{2} + 2\pi = \frac{3\pi}{2} \] Now, we calculate the coordinates: \[ x = \cos\left(-\frac{\pi}{2}\right) = 0 \] \[ y = \sin\left(-\frac{\pi}{2}\right) = -1 \] Thus, the terminal point for \( t = -\frac{\pi}{2} \) is \((0, -1)\). ### Summary of Terminal Points - For \( t = 3\pi \): \((-1, 0)\) - For \( t = -\pi \): \((-1, 0)\) - For \( t = -\frac{\pi}{2} \): \((0, -1)\) These are the terminal points on the unit circle for the given angles.