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Cartesian equations are expressions that relate coordinates in the Cartesian coordinate system, using the familiar axes of x, y, and z. This system is often used in algebra, geometry, and calculus for its straightforward depiction of space and shapes. Unlike cylindrical coordinates, which involve radial distances and angles, Cartesian coordinates directly relate to positions along perpendicular axes.

For example, let's consider the conversion of a cylindrical coordinate equation to a Cartesian equation. In problem (a), we have the equation:

• Given in cylindrical coordinates: \( r^{2} + z^{2} = 9 \)
• Conversion involves recognizing that \( r^{2} \) can be substituted as \( x^{2} + y^{2} \) since in cylindrical-to-Cartesian conversion, \( r \) is tied to both x and y.

Thus, we arrive at the Cartesian form:

• \( x^2 + y^2 + z^2 = 9 \), depicting a sphere with radius 3 centered at the origin.

Trigonometric identities are mathematical equations that express relationships between the angles and sides of triangles. These identities are essential tools in converting coordinates and simplifying expressions, particularly in problems like the one given in example (c). Here, these identities allow us to manipulate angles directly into linear forms.

For instance, in exercise (c), we are faced with:

• \( r^2 \cos 2\theta + z^2 = 1 \)
• The identity \( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \) is key in transitioning from angular to rectangular representations.

This trigonometric identity enables the expression of \( r^2 \cos 2\theta \) in terms of x and y:

• \( x^2 - y^2 + z^2 = 1 \)

By using these identities, complex forms involving angles are turned into simpler polynomial equations that can be more easily analyzed or graphed.

Coordinate conversion is a process that involves changing the coordinates from one system to another, for example, converting from cylindrical to Cartesian coordinates. Each coordinate system has its strengths, and conversion often reveals different insights about the geometry or properties of a spatial figure.

In cylindrical coordinates, a point in space is defined by three parameters: the radial distance \( r \), the angle \( \theta \) from the positive x-axis, and the height z (same as in Cartesian). To convert these to Cartesian coordinates:

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

Using these relationships, any equation or point described in cylindrical coordinates can be re-expressed in the linear and familiar Cartesian framework. For example, converting \( r^2 \cos^2 \theta + z^2 = 4 \) from the exercise (b) is achieved by recognizing:

• \( r^2 \cos^2 \theta = x^2 \)
• Resulting in \( x^2 + z^2 = 4 \)

This converted equation describes a cylinder with a radius of 2 centered along the y-axis. Mastery of coordinate conversion allows deeper understanding and manipulation of mathematical and physical problems.