Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Determining Point Coordinates on a Circle
To find the coordinates of a point on a circle with a radius of 10 units corresponding to an angle of 300 degrees, use the following formulae for converting polar coordinates to Cartesian coordinates:
\[ x = r \cdot \cos(\theta) \]
\[ y = r \cdot \sin(\theta) \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the angle in degrees, given as 300° in this case.
Given:
- Radius (\( r \)) = 10 units
- Angle (\( \theta \)) = 300°
First, convert the angle from degrees to radians since the trigonometric functions in standard calculators and most software use radians.
\[ \theta_{radians} = 300° \times \left( \frac{\pi}{180} \right) = \frac{300\pi}{180} = \frac{5\pi}{3} \]
Now, calculate the \( x \) and \( y \) coordinates using the cosine and sine functions respectively:
\[ x = 10 \cdot \cos\left( \frac{5\pi}{3} \right) \]
\[ y = 10 \cdot \sin\left( \frac{5\pi}{3} \right) \]
Using a calculator or trigonometric tables:
\[ \cos\left( \frac{5\pi}{3} \right) = \frac{1}{2} \]
\[ x = 10 \cdot \frac{1}{2} = 5 \]
\[ \sin\left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2} \]
\[ y = 10 \cdot \left( -\frac{\sqrt{3}}{2} \right) = -5\sqrt{3} \]
Thus,
\[ y = -8.660 \] (since \(\sqrt{3} \approx 1.732\))
Hence, the coordinates are:
\[ (x, y) = (5, -8.660) \]
**Answer:**
Round your answers to three decimal places:
\[ (x, y) = (5.000, -8.660) \]
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Transcribed Image Text:---
### Determining Point Coordinates on a Circle
To find the coordinates of a point on a circle with a radius of 10 units corresponding to an angle of 300 degrees, use the following formulae for converting polar coordinates to Cartesian coordinates:
\[ x = r \cdot \cos(\theta) \]
\[ y = r \cdot \sin(\theta) \]
Where:
- \( r \) is the radius of the circle.
- \( \theta \) is the angle in degrees, given as 300° in this case.
Given:
- Radius (\( r \)) = 10 units
- Angle (\( \theta \)) = 300°
First, convert the angle from degrees to radians since the trigonometric functions in standard calculators and most software use radians.
\[ \theta_{radians} = 300° \times \left( \frac{\pi}{180} \right) = \frac{300\pi}{180} = \frac{5\pi}{3} \]
Now, calculate the \( x \) and \( y \) coordinates using the cosine and sine functions respectively:
\[ x = 10 \cdot \cos\left( \frac{5\pi}{3} \right) \]
\[ y = 10 \cdot \sin\left( \frac{5\pi}{3} \right) \]
Using a calculator or trigonometric tables:
\[ \cos\left( \frac{5\pi}{3} \right) = \frac{1}{2} \]
\[ x = 10 \cdot \frac{1}{2} = 5 \]
\[ \sin\left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2} \]
\[ y = 10 \cdot \left( -\frac{\sqrt{3}}{2} \right) = -5\sqrt{3} \]
Thus,
\[ y = -8.660 \] (since \(\sqrt{3} \approx 1.732\))
Hence, the coordinates are:
\[ (x, y) = (5, -8.660) \]
**Answer:**
Round your answers to three decimal places:
\[ (x, y) = (5.000, -8.660) \]
---
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