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Polar coordinates provide a way to locate points in a plane using a distance and an angle. Imagine a circle with a radius extending from its center. In this system, the origin of the circle is the point of reference. Here are some key aspects of polar coordinates:

• Radius (r): This is the distance from the origin to the point in question. In a polar equation like \( r = 7 \), the radius is constantly 7.
• Angle (θ): This is measured from the positive x-axis. It's often measured in degrees or radians. The angle helps specify the direction of the radius.

Polar coordinates are particularly useful when dealing with scenarios involving rotations or circular paths. In our given exercise, the equation \( r = 7 \) describes a circle where all points are 7 units away from the origin, regardless of the angle.

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes to describe a location on a plane. This system uses an x-coordinate and a y-coordinate to indicate positions. Here's what you need to know about rectangular coordinates:

• x-coordinate: This is the horizontal distance from the y-axis. The x-coordinate is calculated using \( x = r \cos \theta \).
• y-coordinate: This represents the vertical distance from the x-axis. It is found using \( y = r \sin \theta \).

By converting from polar to rectangular coordinates, you are essentially re-mapping a point based on its distance and angle to a point using direct horizontal and vertical distances. This helps in graphing and visualizing mathematical relationships more easily. For the example of \( r = 7 \), conversion provides the points along the circle defined by \( x^2 + y^2 = 49 \), showing how those polar coordinates spread across the Cartesian grid.

The equation of a circle in rectangular coordinates is given by \( x^2 + y^2 = R^2 \), where \( R \) is the radius of the circle. In our exercise, we start with the polar equation \( r = 7 \). After converting to rectangular coordinates, we derive:

• \( x = 7 \cos \theta \)
• \( y = 7 \sin \theta \)

By squaring these and adding them, we arrive at the rectangular form of the circle's equation: \( x^2 + y^2 = 49 \). This represents a circle centered at the origin with a radius of 7 units. It shows that every point (x, y) lies on a circular path equidistant from the origin by 7 units. Recognizing this equation in geometric contexts confirms that you're dealing with a perfect circle, which is a fundamental concept in both algebra and geometry.