Question

Plot each point in polar coordinates. $$\left(2.5,215^{\circ}\right)$$

Short answer

To plot the point \( (2.5, 215^{\circ}) \), measure an angle of \( 215^{\circ} \) counterclockwise from the positive x-axis and from that angle, move outward to a distance of 2.5 units.

Step-by-step solution

Understand Polar Coordinates

Polar coordinates are expressed as a pair \( (r, \theta) \) where \( r \) is the radius or the distance from the origin, and \( \theta \) is the angle in degrees or radians measured counterclockwise from the positive x-axis. For the given point \( (2.5, 215^{\circ}) \) the radius \( r = 2.5 \) and the angle \( \theta = 215^{\circ} \) in degrees.

Convert Angle to Radians (Optional)

For plotting, you may convert the angle from degrees to radians to use certain calculators or software that default to radians. However, this step is optional since the polar plot can be made using degrees. To convert degrees to radians, use the formula \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \) if needed.

Plot the Point

On a polar grid, start at the origin (0,0). Move counterclockwise from the positive x-axis to an angle of \( 215^{\circ} \) or the corresponding radian measure if converted. From there, move outward radially to a distance of 2.5 units. Mark the point at this location.

Key concepts

Plotting Polar Coordinates

Plotting polar coordinates involves representing points in a two-dimensional plane using a combination of a radius and an angle rather than the traditional Cartesian coordinates of x and y.

To plot a polar coordinate, imagine a pole (the origin) and a polar axis (similar to the x-axis in Cartesian coordinates). From the pole, move along the polar axis an angle \( \theta \) measured in degrees or radians, towards the direction where the angle is specified. Once you have the direction, move away from the pole along this line a distance equal to the radius \( r \) of the coordinate.

Example: Plotting (2.5, 215°)

For the example point \( (2.5, 215^\circ) \) provided in the exercise, start at the origin (0,0). Rotate from the positive x-axis (polar axis) counterclockwise until you reach a 215° angle. From there, measure a distance of 2.5 units radially from the origin and mark the point.

Remember, regardless of the plotting process, the key to mastering polar coordinates is to maintain a clear image of the polar grid in your mind and understand the roles of radius and angle.

Converting Degrees to Radians

Understanding the relationship between degrees and radians is crucial when dealing with trigonometric functions or plotting in different contexts, as radians are often the default in mathematics and software tools.

To convert an angle from degrees to radians, multiply the number of degrees by \( \frac{\pi}{180} \). The conversion factor \( \frac{\pi}{180} \) represents the ratio of a full circle in radians (\(\pi\) radians is equivalent to 180°).

Applying the Conversion Formula

For instance, to convert 215° to radians, calculate \( 215^\circ \times \frac{\pi}{180} \). This conversion simplifies the manipulation of angles in various mathematical problems and allows for the integration of angles into more complex equations, which typically expect radian measurements.

Understanding Radius and Angle

The fundamental components of polar coordinates are the radius \( r \) and the angle \( \theta \). The radius represents the distance from the origin to the point, while the angle provides direction from the reference line (positive x-axis).

The radius can be any positive number or zero (where the point would fall at the origin). As for the angle, it is typically measured in degrees or radians from the positive x-axis, moving counterclockwise for positive angles and clockwise for negative angles.

Radius-Angle Relationship

The distance (radius) and direction (angle) are interdependent; changing either value will change the point's position. For example, if two points have the same angle but different radii, they will lie on the same line from the origin but at different distances; whereas if they have the same radius but different angles, they will be the same distance from the origin but in different directions.