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. (3, 90°)

Verified step by step guidance

1. Understand that polar coordinates are given in the form $(r,\theta )$, where $r$ is the radial distance from the origin and $\theta$ is the angle from the positive x-axis.
2. Identify the given polar coordinates: $(3,{90}^{\circ })$. Here, $r=3$ and $\theta ={90}^{\circ }$.
3. Recognize that an angle of ${90}^{\circ }$ corresponds to the positive y-axis in the polar coordinate system.
4. Plot the point by moving 3 units from the origin along the line that makes a ${90}^{\circ }$ angle with the positive x-axis.
5. Mark the point on the polar coordinate system at the intersection of the circle with radius 3 and the line at ${90}^{\circ }$.

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 circular or rotational patterns.

Plotting Points in Polar Coordinates

To plot a point given in polar coordinates, first identify the radial distance 'r' and the angle 'θ'. The angle is measured counterclockwise from the positive x-axis. For example, the point (3, 90°) means to move 3 units away from the origin at an angle of 90°, which points directly upward along the y-axis.

Conversion Between Polar and Cartesian Coordinates

Understanding how to convert between polar and Cartesian coordinates is essential. The conversion formulas are x = r * cos(θ) and y = r * sin(θ). This allows for a comprehensive understanding of the point's location in both coordinate systems, facilitating easier plotting and analysis.