In Exercises 1–10, indicate if the point with the given polar coordinates is represented by A, B, C, or D on the graph. (−3, 5π/4)

Verified step by step guidance

1. Step 1: Understand polar coordinates. Polar coordinates are given in the form (r, θ), where r is the radius (distance from the origin) and θ is the angle measured from the positive x-axis.
2. Step 2: Analyze the given coordinates. Here, r = -3 and θ = $\frac{5\pi }{4}$. The negative radius indicates that the point is in the opposite direction of the angle.
3. Step 3: Determine the angle direction. The angle $\frac{5\pi }{4}$ is in the third quadrant, as it is more than $\pi$ (180 degrees) but less than $\frac{3\pi }{2}$ (270 degrees).
4. Step 4: Adjust for the negative radius. Since the radius is negative, the point is actually in the opposite direction of the angle, which places it in the first quadrant.
5. Step 5: Identify the point on the graph. Based on the adjusted position, locate the point in the first quadrant that corresponds to the given polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a plane using a distance from a reference point (the origin) and an angle from a reference direction (usually the positive x-axis). The first value indicates the radius (distance from the origin), while the second value is the angle in radians. For example, the point (−3, 5π/4) means moving 3 units in the direction of the angle 5π/4, which is in the third quadrant.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians, with radians being the standard unit in mathematics. One full rotation (360 degrees) is equivalent to 2π radians. The angle 5π/4 radians corresponds to 225 degrees, indicating a direction that points diagonally in the third quadrant of the polar coordinate system.

Negative Radius in Polar Coordinates

A negative radius in polar coordinates indicates that the point is located in the opposite direction of the angle specified. For instance, a point with coordinates (−3, 5π/4) means to move 3 units in the direction opposite to 5π/4, effectively placing the point in the second quadrant. This concept is crucial for accurately determining the location of points in polar graphs.