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, −3π/4)

Verified step by step guidance

1. Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis in the counterclockwise direction.
2. The given polar coordinates are \((-3, -\frac{3\pi}{4})\). Here, \(r = -3\) and \(\theta = -\frac{3\pi}{4}\).
3. A negative radius means that the point is in the opposite direction of the angle. So, instead of moving \(-3\) units in the direction of \(-\frac{3\pi}{4}\), move \(3\) units in the direction of \(\frac{\pi}{4}\) (which is the opposite direction).
4. The angle \(-\frac{3\pi}{4}\) is equivalent to \(\frac{5\pi}{4}\) when measured in the standard counterclockwise direction. However, since we are considering the opposite direction due to the negative radius, we use \(\frac{\pi}{4}\).
5. Now, locate the point by moving \(3\) units from the origin in the direction of \(\frac{\pi}{4}\). This will help you identify which point (A, B, C, or D) corresponds to the given polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a two-dimensional space using a distance from a reference point (the origin) 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 radians. Understanding how to interpret these coordinates is essential for locating points on a polar graph.

Angle Measurement in Radians

In trigonometry, angles can be measured in degrees or radians. Radians are a more natural way to measure angles in the context of circles, where one full rotation (360 degrees) is equivalent to 2π radians. The angle θ = -3π/4 indicates a clockwise rotation from the positive x-axis, which is crucial for determining the correct position of the point in 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 example, a point with coordinates (−3, −3π/4) means to move 3 units in the direction opposite to the angle of −3π/4. This concept is vital for accurately plotting points on a polar graph, as it affects the quadrant in which the point lies.