Problem 24 - Precalculus Mathematics for Calculus

Question

Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$t=\frac{3 \pi}{2}$$

Step-by-Step Solution

Answer

The terminal point is \((0, -1)\).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin \((0, 0)\). Any angle \(t\), measured in radians, places a corresponding point \(P(x,y)\) on the circle.

Step 2: Determine the Quadrant

To locate the terminal point, we need to determine which quadrant \(t = \frac{3\pi}{2}\) falls into. Angles start at \(0\) on the positive x-axis and move counterclockwise. \(\pi/2\) corresponds to the top of the circle, \(\pi\) corresponds to the left, and \(3\pi/2\) corresponds to the bottom of the circle.

Step 3: Locate the Point on the Circle

The angle \(t = \frac{3\pi}{2}\) is directly at the bottom of the unit circle, right on the negative y-axis. This corresponds to the point \((0, -1)\) because at this angle, the x-coordinate is 0 and the y-coordinate is -1.

Key Concepts

Angles in Radians

Understanding angles in radians is essential when working with the unit circle. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of the circle. One radian is the angle formed when the radius is wrapped along the edge of the circle. Hence, the circumference of a full circle is equivalent to about 6.28 radians, or precisely, \(2\pi\) radians.

• \(\pi\) radians equal 180 degrees, which is half a circle.
• \(\frac{\pi}{2}\) radians represent 90 degrees, or a quarter of the circle.
• \(\frac{3\pi}{2}\) radians, as in our problem, denote three-quarters of the way around the circle, landing at the bottom of the unit circle.

Radians offer a convenient way to measure angles, especially in trigonometric contexts, as they directly relate angles to the radius and the arc length.

Terminal Point

The terminal point on the unit circle is the ending spot of an angle's rotation, starting from the positive x-axis. It is where the angle halts on the circle.
When \(t = \frac{3\pi}{2}\), our terminal point is where the angle has swept three-quarters of the circle, ending at the negative y-axis. This specific rotation results in a point positioned directly at the bottom of the circle.

• For \(\frac{3\pi}{2}\), we use the knowledge of quadrant positions and circle orientation to find the terminal point.
• It lands on the coordinate \((0, -1)\), indicating the basics of trigonometric angles and points within a circle.

This method helps us visualize and calculate points based on given angles effortlessly.

Coordinate System

The coordinate system is a cornerstone of understanding geometry and trigonometry. In the unit circle context, it defines how points are located based on x and y coordinates combined with a radius of 1.

• The origin is found at \((0,0)\), the center of the unit circle.
• Each point on the circle has coordinates \((x, y)\) that depend on the angle \(t\).
• The x-coordinate represents the horizontal displacement, while the y-coordinate represents the vertical displacement from the origin.

In the problem, for an angle of \(\frac{3\pi}{2}\), the coordinate is located at \((0, -1)\), fully on the y-axis, with no horizontal movement from the center. This represents the use of coordinates to precisely define positions on geometric shapes.

Trigonometric Functions

Trigonometric functions are fundamental in connecting angles with their corresponding coordinates on the unit circle. These functions define relationships between the angles and side lengths in right triangles, extended to circles.

• Sine function corresponds to the y-coordinate on the unit circle.
• Cosine function correlates with the x-coordinate.
• Tangent is the ratio of sine over cosine.

For the angle \(t = \frac{3\pi}{2}\), using trigonometric functions helps identify and predict the location of the terminal point.

• The sine of \(\frac{3\pi}{2}\) equals \(-1\), reflecting the y-coordinate.
• The cosine is \(0\), showcasing no x-displacement.

These trigonometric insights empower us to determine where the terminal point lies and establish the physical shape on the unit circle with ease.