Problem 128 - Precalculus

Question

For the following exercises, find the requested value. Find the coordinates of the point on a circle with radius 15 corresponding to an angle of \(220^{\circ} .\)

Step-by-Step Solution

Answer

The coordinates are approximately (-11.49, -9.64).

Step 1: Convert the angle from degrees to radians

To find the coordinates on a circle, we often convert the angle in degrees to radians. The formula to convert degrees to radians is \( \text{radians} = \frac{\pi}{180} \times \text{degrees} \). Substituting 220° into the formula gives us: \( 220^{\circ} \times \frac{\pi}{180} = \frac{11\pi}{9} \) radians.

Step 2: Use the unit circle formulas

To find the coordinates \((x, y)\) on a circle, we use the formulas: \( x = r \cdot \cos(\theta) \) and \( y = r \cdot \sin(\theta) \), where \( r \) is the radius and \( \theta \) is the angle in radians. Here, \( r = 15 \) and \( \theta = \frac{11\pi}{9} \).

Step 3: Calculate the x-coordinate

Using the formula \( x = r \cdot \cos(\theta) \), substitute \( r = 15 \) and \( \theta = \frac{11\pi}{9} \). Thus, \( x = 15 \cdot \cos(\frac{11\pi}{9}) \). Calculating this gives us \( x \approx -11.49 \).

Step 4: Calculate the y-coordinate

Using the formula \( y = r \cdot \sin(\theta) \), substitute \( r = 15 \) and \( \theta = \frac{11\pi}{9} \). Thus, \( y = 15 \cdot \sin(\frac{11\pi}{9}) \). Calculating this gives us \( y \approx -9.64 \).

Step 5: Write the coordinates

The coordinates of the point on the circle are found by placing the calculated x and y values together, forming the coordinates \((-11.49, -9.64)\).

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Imagine a circle centered at the origin of a coordinate system with a radius of exactly 1. This circle helps students understand the relationships between angles and distances.
The unit circle is incredibly handy for visualizing trigonometric functions like sine and cosine.

• Each point on the unit circle represents coordinates \((\cos(\theta), \sin(\theta))\), where \(\theta\) is the angle formed with the positive x-axis.
• The x-coordinate gives the cosine of the angle, while the y-coordinate provides the sine.

If you stretch or shrink the radius, it doesn’t change the sine and cosine values because ratios are maintained. The same principle helps in solving the problem of finding a point on any circle using trigonometry.

Radians to Degrees Conversion

Angles can be measured in different units, and two of them are degrees and radians. Understanding radians is crucial for tasks involving trigonometry, especially in advanced mathematics and physics.
The conversion between degrees and radians is very straightforward:

• To convert from degrees to radians, use the formula \(\text{radians} = \frac{\pi}{180} \times \text{degrees}\).
• For example, 1 degree equals \(\frac{\pi}{180}\,\text{radians}\).
• To convert from radians to degrees, you would do the reverse by multiplying the radians by \(\frac{180}{\pi}\).

In this exercise, converting 220 degrees into radians was crucial to accurately find the coordinates on the circle.

Coordinates of a Point on a Circle

To find a point on a circle, you need three things: the radius of the circle, the angle you're using, and the origin of the circle grid. Usually, the circle is centered at \((0, 0)\) like the unit circle.
For a circle of any radius, like 15 in the provided example, the formulas to find the coordinates become:

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

In this context, the formulas yield the exact point on the circle by only plugging in the radius and the angle in radians. This approach allows you to determine the specific location of any angle on a circle of any radius.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are foundational in the study of triangles and circles. They describe the relationships between the angles and sides of triangles, especially right triangles.
On the unit circle:

• \(\cos(\theta)\) represents the x-coordinate or the horizontal position of a point on the circle.
• \(\sin(\theta)\) gives the y-coordinate or the vertical position.
• These functions allow you to transform angles into spatial coordinates directly.

Understanding how these functions apply is important for tasks such as the exercise, helping visualize the circle’s points and understand the trigonometric relationships in a tangible way.