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. <Step 1: Understand Polar Coordinates> 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. <Step 2: Analyze the Given Coordinates> The given polar coordinates are \((-3, -\frac{3\pi}{4})\). Here, \(r = -3\) and \(\theta = -\frac{3\pi}{4}\).
3. <Step 3: Interpret the Negative Radius> 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. <Step 4: Convert the Angle> 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. <Step 5: Locate the Point> 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.