Problem 76 - Precalculus

Question

What do the \(x\) - and \(y\) -coordinates of the points on the unit circle represent?

Step-by-Step Solution

Answer

The \\(x\\)-coordinate represents \\(\cos(\theta)\\) and the \\(y\\)-coordinate represents \\(\sin(\theta)\\).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. Any point on this circle can be described as \(x, y\), where \(x\) and \(y\) are the coordinates of the point.

Step 2: Identify the Connection to Trigonometric Functions

In the unit circle, the angle \( \theta \) measured from the positive x-axis to a point on the circle is used to determine the coordinates of the point. The \(x\)-coordinate represents the cosine of the angle, \(x = \cos(\theta)\), and the \(y\)-coordinate represents the sine of the angle, \(y = \sin(\theta)\).

Step 3: Summarize the Meaning of Coordinates

Each point \( (x, y) \) on the unit circle, corresponds to \( (\cos(\theta), \sin(\theta)) \), where \( \theta \) is the angle measured in radians from the positive x-axis. These trigonometric functions describe the horizontal and vertical displacements from the origin to that point.

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, especially in relation to angles and circles. They help describe relationships between the angles and sides of triangles, particularly right-angled triangles. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).

• Sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• Tangent of an angle is the ratio of the sine to the cosine, or the opposite side to the adjacent side.

These functions are crucial to various fields such as physics, engineering, and computer science. They also extend beyond triangles through the use of the unit circle, providing a broader way to analyze angles.

Coordinates on the Unit Circle

The unit circle is a powerful tool in mathematics for understanding trigonometric functions. It is a simple circle with a radius of 1, centered at the origin (0,0) of a coordinate plane. Points on this circle can be represented by coordinates (\(x, y\)).

• The x-coordinate of a point on the unit circle is given by \(\cos(\theta)\)
• The y-coordinate is given by \(\sin(\theta)\)

The angle \(\theta\) is measured in radians from the positive x-axis to the radius drawn to the point on the circle. These coordinates reflect the values of trigonometric functions for different angles, offering a visual and mathematical representation of these concepts.

Cosine and Sine Relationship

The relationship between cosine and sine is essential for understanding the properties of the unit circle and trigonometric functions. On the unit circle, each point’s coordinates correlate with an angle \(\theta\).
Cosine (\(\cos(\theta)\)) represents the x-coordinate, showing how far left or right the point is from the origin. Meanwhile, sine (\(\sin(\theta)\)) represents the y-coordinate, indicating how far up or down the point is.
These coordinates and trigonometric functions help describe angles perfectly. Importantly, they confirm that for any angle, the values of cosine and sine will always be between -1 and 1 because they are projections within the bounds of the unit circle.
This relationship is simple yet profound, hinting at deeper concepts such as periodicity and symmetry in trigonometry.