Problem 24 - Algebra and Trigonometry

Question

23-32 \(\approx\) 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 of the coordinate plane (0,0). The "terminal point" refers to the coordinates on the unit circle at a specific angle, or value of \(t\). Since the circle has a radius of 1, any point \((X, Y)\) on the circle satisfies the equation \(X^2 + Y^2 = 1\).

Step 2: Locating the Angle on the Unit Circle

The given value of \(t = \frac{3\pi}{2}\) is the angle in radians. This angle measures counterclockwise from the positive x-axis. \(\frac{3\pi}{2}\) radians is equivalent to 270 degrees, which is three quarters of the way around the circle, placing it along the negative y-axis.

Step 3: Coordinate of Terminal Point

At \(\frac{3\pi}{2}\) radians, the terminal point is at the bottom of the unit circle. Therefore, the x-coordinate of the point is 0, and since it is on the negative y-axis, the y-coordinate is -1. Thus, the terminal point \(P(X, Y)\) is \((0, -1)\).

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. However, in this context, it also deals with circular functions and angles measured in a coordinate system.

In trigonometry, the unit circle is a fundamental concept. A unit circle has a radius of exactly 1 and is centered at the origin (0,0) on a coordinate plane. The main purpose of the unit circle is to define trigonometric functions for all real numbers. When a real number, represented by angle t, rotates from the positive x-axis around this circle, it helps in determining the sine and cosine of that angle:

• The x-coordinate on the unit circle corresponds to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.

Understanding this connection is essential for solving various trigonometric problems and identifying coordinates of points on the circle.

Exploring Radians

In trigonometry, radians are a way to measure angles based on the radius of a circle. Unlike degrees, where a full circle is 360 degrees, in radian measure, a full circle is 2π radians. This creates a direct link between the angle and the circle's arc.

Radians simplify many mathematical expressions and are the standard unit of angular measure in mathematics. For example, an angle of:

• π radians corresponds to 180 degrees, forming a straight line.
• 2π radians corresponds to 360 degrees, completing a full rotation.
• π/2 radians equates to 90 degrees, forming a right angle.
• 3π/2 radians, used in the exercise, equals 270 degrees.

Being comfortable with radians is crucial, as they are used extensively in advanced math problems and in describing the positions of points along the unit circle.

Navigating Coordinates

Coordinates are used to precisely determine the position of a point in space on a plane. On a two-dimensional plane, this is done using an x-coordinate and a y-coordinate.

In the context of the unit circle, every point on the circle can be described by these (x, y) coordinates:

• The x-coordinate represents the horizontal distance from the origin.
• The y-coordinate represents the vertical distance from the origin.

For any angle, the coordinates on the unit circle can be found using trigonometric functions:

• The x-coordinate is given by the cosine of the angle.
• The y-coordinate is given by the sine of the angle.

In exercises like identifying a terminal point on the unit circle, accurately determining these coordinates is key to finding the correct answer.

Identifying Terminal Points

Terminal points are specific points on the unit circle that correspond to a given angle. These points are crucial in understanding the geometric representation of angles in trigonometry.

When a point travels around the unit circle starting from the positive x-axis, the endpoint reached is referred to as the terminal point. To find the terminal point for a given angle:

• Convert the angle into radians if it's initially in degrees.
• Determine its position on the unit circle based on the trigonometric standards of sine and cosine.

For example, at 3π/2 radians, the terminal point is located on the negative y-axis. This is because:

• The x-coordinate is 0 (since \(\cos\left( \frac{3\pi}{2} \right) = 0\)").
• The y-coordinate is -1 (since \\(\sin\left( \frac{3\pi}{2} \right) = -1\)).

Understanding how to locate terminal points helps in solving problems involving rotations and periodic functions in trigonometry.