Problem 131 - Precalculus

Question

For the following exercises, find the requested value. Find the coordinates of the point on a circle with radius 16 corresponding to an angle of \(\frac{5 \pi}{9}\) .

Step-by-Step Solution

Answer

Coordinates: \((-15.035, 5.472)\).

Step 1: Understanding the Problem

We are tasked with finding the coordinates of a point on a circle. The circle has a radius of 16 and the point corresponds to the angle \( \frac{5\pi}{9} \) radians.

Step 2: Convert Angle to Cartesian Coordinates

Since the point lies on the circle, we can use the parametric equations for a circle to find its coordinates:\[ x = r \cdot \cos(\theta) \]\[ y = r \cdot \sin(\theta) \]where \( r = 16 \) and \( \theta = \frac{5\pi}{9} \).

Step 3: Calculate the X-coordinate

Use the cosine function to find the x-coordinate of the point:\[ x = 16 \cdot \cos\left(\frac{5\pi}{9}\right) \]

Step 4: Calculate the Y-coordinate

Use the sine function to find the y-coordinate of the point:\[ y = 16 \cdot \sin\left(\frac{5\pi}{9}\right) \]

Step 5: Evaluate the Trigonometric Functions

Evaluate \( \cos\left(\frac{5\pi}{9}\right) \) and \( \sin\left(\frac{5\pi}{9}\right) \) using a calculator or trigonometric table, and multiply by 16:\[ x = 16 \cdot (-0.93969) \approx -15.035 \]\[ y = 16 \cdot (0.34202) \approx 5.472 \]

Step 6: State the Result

The coordinates of the point on the circle corresponding to the angle \( \frac{5\pi}{9} \) are approximately \((-15.035, 5.472)\).

Key Concepts

coordinates on a circle

When talking about the coordinates on a circle, we're often referring to the position of a point on the circumference relative to the center. In a simple Cartesian plane, if we know a point's coordinates, we can determine its position in terms of an

• “x” coordinate, which describes how far left or right the point is from the origin, and
• “y” coordinate, which describes how far up or down the point is from the origin.

For a circle with a given radius, each point on its boundary can be described using trigonometric functions. This involves knowing the angle that the radius forms with a horizontal line, often measured in radians.
The positive "x" axis is the horizontal line from the origin, pointing to the right. The coordinates are calculated using:

• the cosine function for the "x" coordinate, and
• the sine function for the "y" coordinate.

Each coordinate is essentially a scaled version of these trigonometric values, multiplied by the radius of the circle. For instance, with a radius of 16, every point on the circle can be found by multiplying the circle's radius with the cosine and sine of the angle in question.

angle conversion to radians

In trigonometry, angles can be measured in two ways: degrees and radians. However, radians are the standard unit of angular measure used in many fields of mathematics. To convert an angle measured in degrees to radians, we use the formula: \[\text{radians} = \text{degrees} \times \left(\frac{\pi}{180}\right)\]This conversion is crucial when working with trigonometric functions because these functions inherently use radians in their calculations.
You can think of radians as a way of linking the radius of a circle to the length of an arc. One complete revolution around a circle is equal to \2\pi radians, which corresponds to 360 degrees.

For example, an angle of 90 degrees, when converted to radians, becomes:\[90 \times \left(\frac{\pi}{180}\right) = \frac{\pi}{2}\]Understanding this conversion aids in computing exact values of trigonometric functions at different angles, thus helping us find points on the circle using their parametric equations.

parametric equations for a circle

Parametric equations allow us to express the coordinates of the points on a curve, such as a circle, as functions of a parameter, often the angle \(\theta\) in radians. To describe a circle with radius \(r\) centered at the origin in terms of parametric equations, we use:

• \(x = r \cdot \cos\theta\)
• \(y = r \cdot \sin\theta\)

These equations take advantage of the circular functions sine and cosine to map out the entire circle's outline. By varying \(\theta\), which usually ranges from \(0\) to \(2\pi\), we can trace out every point along the circle's edge.
The parametric form is particularly useful because it relates angles measured from the positive x-axis directly to the circle's radius. This eliminates the need for constructing the circle manually or relying on angular conversion if already in radians.
These equations are versatile and apply to any circle, regardless of its size, as long as we adjust the \(r\) value accordingly.