Change the equation from rectangular form to polar form. 4x² - y' = 4
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**Problem 5. (a)**
**Objective:** Change the equation from rectangular form to polar form.
**Given Equation:**
\[4x^2 - y^2 = 4\]
**Steps for Conversion:**
1. **Introduce Polar Coordinates:**
- In polar coordinates, the rectangular coordinates \(x\) and \(y\) can be expressed in terms of \(r\) (the radius) and \(\theta\) (the angle) as follows:
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
2. **Substitute Polar Coordinates into the Given Equation:**
- Replace \(x\) and \(y\) in the given equation with the polar coordinate expressions:
\[ 4(r \cos \theta)^2 - (r \sin \theta)^2 = 4 \]
3. **Simplify the Equation:**
- Expand and combine like terms:
\[ 4r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4 \]
- Factor out \(r^2\) from the left side of the equation:
\[ r^2 (4 \cos^2 \theta - \sin^2 \theta) = 4 \]
4. **Solve for \(r^2\):**
- Isolate \(r^2\) by dividing both sides by \( (4 \cos^2 \theta - \sin^2 \theta) \):
\[ r^2 = \frac{4}{4 \cos^2 \theta - \sin^2 \theta} \]
**Final Polar Form:**
\[ r^2 = \frac{4}{4 \cos^2 \theta - \sin^2 \theta} \]
This completes the conversion of the given rectangular equation to its polar form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39f0d4ee-d4a6-4902-9c76-b777b279390d%2F376ab4bf-aa4d-4881-ad61-98e323be900d%2Fz0vhgug.png&w=3840&q=75)
Transcribed Image Text:**Problem 5. (a)**
**Objective:** Change the equation from rectangular form to polar form.
**Given Equation:**
\[4x^2 - y^2 = 4\]
**Steps for Conversion:**
1. **Introduce Polar Coordinates:**
- In polar coordinates, the rectangular coordinates \(x\) and \(y\) can be expressed in terms of \(r\) (the radius) and \(\theta\) (the angle) as follows:
\[ x = r \cos \theta \]
\[ y = r \sin \theta \]
2. **Substitute Polar Coordinates into the Given Equation:**
- Replace \(x\) and \(y\) in the given equation with the polar coordinate expressions:
\[ 4(r \cos \theta)^2 - (r \sin \theta)^2 = 4 \]
3. **Simplify the Equation:**
- Expand and combine like terms:
\[ 4r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4 \]
- Factor out \(r^2\) from the left side of the equation:
\[ r^2 (4 \cos^2 \theta - \sin^2 \theta) = 4 \]
4. **Solve for \(r^2\):**
- Isolate \(r^2\) by dividing both sides by \( (4 \cos^2 \theta - \sin^2 \theta) \):
\[ r^2 = \frac{4}{4 \cos^2 \theta - \sin^2 \theta} \]
**Final Polar Form:**
\[ r^2 = \frac{4}{4 \cos^2 \theta - \sin^2 \theta} \]
This completes the conversion of the given rectangular equation to its polar form.
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