To convert the given Cartesian equation [tex]\( x^2 - y^2 = 4 \)[/tex] into polar coordinates, follow the steps below:
1. Recall the relationships between Cartesian and Polar coordinates:
- [tex]\( x = r \cos \theta \)[/tex]
- [tex]\( y = r \sin \theta \)[/tex]
2. Substitute these relationships into the given Cartesian equation:
- Substitute [tex]\( x = r \cos \theta \)[/tex] and [tex]\( y = r \sin \theta \)[/tex] into [tex]\( x^2 - y^2 = 4 \)[/tex].
[tex]\[
(r \cos \theta)^2 - (r \sin \theta)^2 = 4
\][/tex]
3. Simplify the left side of the equation:
- Expand the terms:
[tex]\[
r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4
\][/tex]
- Factor out the common term [tex]\( r^2 \)[/tex]:
[tex]\[
r^2 (\cos^2 \theta - \sin^2 \theta) = 4
\][/tex]
4. Use a trigonometric identity:
- Recognize that [tex]\( \cos^2 \theta - \sin^2 \theta \)[/tex] is the trigonometric identity for [tex]\( \cos(2\theta) \)[/tex]:
[tex]\[
r^2 \cos(2\theta) = 4
\][/tex]
Thus, the equivalent polar equation for [tex]\( x^2 - y^2 = 4 \)[/tex] is [tex]\( r^2 \cos(2\theta) = 4 \)[/tex].
The correct answer from the provided choices is:
D
