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, −135°)

Verified step by step guidance

1. Convert the polar coordinates (3, -135°) to Cartesian coordinates using the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
2. Substitute \( r = 3 \) and \( \theta = -135° \) into the formulas: \( x = 3 \cos(-135°) \) and \( y = 3 \sin(-135°) \).
3. Recall that \( \cos(-135°) = -\cos(135°) \) and \( \sin(-135°) = -\sin(135°) \) because cosine is an even function and sine is an odd function.
4. Calculate \( \cos(135°) \) and \( \sin(135°) \) using the unit circle or known values: \( \cos(135°) = -\frac{\sqrt{2}}{2} \) and \( \sin(135°) = \frac{\sqrt{2}}{2} \).
5. Substitute these values back into the equations to find the Cartesian coordinates: \( x = 3 \times -\frac{\sqrt{2}}{2} \) and \( y = 3 \times -\frac{\sqrt{2}}{2} \).

Key Concepts

Polar Coordinates

Polar coordinates represent points 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 degrees or radians. Understanding how to interpret these coordinates is essential for locating points on a polar graph.

Angle Measurement

Angles in polar coordinates can be measured in degrees or radians, with positive angles typically measured counterclockwise from the positive x-axis. A negative angle, such as -135°, indicates a clockwise rotation. This understanding is crucial for accurately plotting points and determining their positions relative to the axes.

Graphing Polar Coordinates

To graph polar coordinates, one must convert the polar point into Cartesian coordinates or directly plot it on a polar grid. The point (3, -135°) means moving 3 units from the origin at an angle of -135°, which corresponds to a specific location in the polar plane. Familiarity with the polar grid and how to interpret these coordinates visually is key to solving the problem.