Question

Plot \(\left(4, \frac{5 \pi}{3}\right)\) and \(\left(-3,-\frac{7 \pi}{2}\right)\) on the polar plane.

Short answer

Plot point (4, 60°) towards 60°, and (-3, 90°) oppositely towards 270° on the polar plane.

Step-by-step solution

Understand the Coordinate System

This exercise involves the polar coordinate system. In polar coordinates, each point is defined by a radius and an angle. The radius indicates the distance from the origin, while the angle represents the direction from the positive x-axis.

Convert Angles to Radians and Locate Them

The points are given using radian measure for the angles:- The angle \(\frac{5\pi}{3}\) radians can be visualized as 300 degrees, as it is greater than \(2\pi\) (or 360 degrees), so we can subtract \(2\pi\) (or 360 degrees), which results in an effective angle of 60 degrees.- The angle \(-\frac{7\pi}{2}\) radians converts to \(-630\) degrees. Adding the equivalent in full circles, typically \(360\) degrees, results in a standardized angle in the range \(0-360\) degrees. Thus, \(-630 + 720 = 90\) degrees.

Plot the First Point

The first point \((4, \frac{5\pi}{3})\) means a radius of 4 units at 60 degrees from the positive x-axis. On the polar plane, locate 60 degrees (equivalent to one and two-thirds of a complete circle) and mark a point 4 units from the center along this direction.

Plot the Second Point

The second point \((-3, -\frac{7\pi}{2})\) involves a negative radius and converts to an angle of 90 degrees after adjustment. With negative radius, the point is in the direction opposite to 90 degrees, which is 270 degrees. Plot a point 3 units away from the origin in the 270-degree direction.

Review and Adjust Points if Necessary

Ensure that both points are situated correctly on the polar plane with respect to their respective angles and account for any radius direction changes due to negative values.

Key concepts

Radian Measure

In the world of mathematics, angles can be measured in various units, with radians being one of the most significant. A radian is an alternative to degrees, providing a more natural measure of angular distance particularly in calculus and trigonometry. The concept of a radian is based on the radius of a circle, with a complete circle containing \(2\pi\) radians.

• One radian is the angle created when the arc length is equal to the radius of the circle.
• A circle's circumference is the distance around it, calculated as \(2\pi\times\)radius. Since 360 degrees is one full revolution, we equate 360 degrees to \(2\pi\) radians.
• Therefore, to convert from degrees to radians, you use the formula: \(\text{radians} = \text{degrees} \times \frac{\pi}{180}\).

Radians provide a neat way to work with angles, especially when dealing with trigonometric functions and calculus, making complex calculations simpler and more intuitive.

Angle Conversion

Angle conversion is a process used to convert between degrees and radians. Understanding both forms of measurement is crucial for plotting polar coordinates accurately.When converting from radians to degrees, you use the relationship \(\pi\) radians being equivalent to 180 degrees. Thus, the conversion formula is:

• Degrees = Radians \(\times \frac{180}{\pi}\).

Example: To convert \(\frac{5\pi}{3}\) radians to degrees:- Use \(\frac{5\pi}{3} \times \frac{180}{\pi} = 300\) degrees.For negative angles:- \(-\frac{7\pi}{2}\) converts to \(-630\) degrees. You can simplify this angle by adding multiples of 360 degrees until it falls between 0 and 360 degrees. Hence, adding 720 degrees gives 90 degrees.This standardization step helps when plotting points, ensuring angles correctly represent direction from the positive x-axis in a polar coordinate system.

Negative Radius

A negative radius is a fascinating aspect of polar coordinates because it changes the way we visualize points. In polar coordinates, a point is typically represented by a positive radius and an angle, but what happens when the radius is negative? In cases where the radius is negative, it suggests that the point is located in the opposite direction of the given angle rather than directly on it. This can appear confusing, but breaking it down simplifies the concept.

• A negative radius means that the direction of the point is reflected about the origin.
• For example, with an angle of 90 degrees and a radius of -3, instead of moving 3 units to the right from the 90-degree direction, you move 3 units left in the 270-degree direction.
Recognizing how to plot points with negative radii helps ensure accuracy when placing the points on a polar plane. This unique feature of polar coordinates adds flexibility and variety to how we perceive and utilize direction and distance.