In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (2, 45°)

Verified step by step guidance

1. Understand that polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis.
2. Identify the given polar coordinates: \((2, 45^\circ)\). Here, \(r = 2\) and \(\theta = 45^\circ\).
3. Locate the angle \(45^\circ\) on the polar coordinate system. This angle is measured counterclockwise from the positive x-axis.
4. From the origin, move along the line at \(45^\circ\) until you reach a distance of 2 units. This is the point \((2, 45^\circ)\) in the polar coordinate system.
5. Plot the point on the polar graph at the intersection of the line at \(45^\circ\) and the circle with radius 2.

Key Concepts

Polar Coordinates

Polar coordinates represent points in a two-dimensional space using a distance from a reference point (the pole) and an angle from a reference direction (usually the positive x-axis). The format is (r, θ), where 'r' is the radial distance and 'θ' is the angle in degrees or radians. This system is particularly useful for representing circular or spiral patterns.

Plotting Points in Polar Coordinates

To plot a point given in polar coordinates, first identify the radial distance 'r' and the angle 'θ'. From the pole, measure the angle 'θ' counterclockwise from the positive x-axis, then move outward from the pole a distance of 'r'. If 'r' is negative, move in the opposite direction along the line defined by the angle.

Conversion Between Polar and Cartesian Coordinates

Understanding how to convert between polar and Cartesian coordinates is essential for interpreting polar points. The conversion formulas are x = r * cos(θ) and y = r * sin(θ). This allows you to translate polar coordinates into the familiar (x, y) format, facilitating easier plotting on a Cartesian grid.