Problem 21 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(3, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The point (3, \( \frac{\pi}{2} \)) is plotted 3 units up along the positive y-axis.

Step 1 - Understand Polar Coordinates

In polar coordinates, a point is represented as \(r, \theta\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis.

Step 2 - Identify the Given Coordinates

The given point in polar coordinates is \(3, \frac{\pi}{2}\). This means the radial distance \(r\) is 3 units, and the angle \(\theta\) is \(\frac{\pi}{2}\) radians.

Step 3 - Interpret the Angle

Convert the angle \(\frac{\pi}{2}\) radians to degrees for simpler interpretation. \(\frac{\pi}{2} = 90^{\circ}\). This means the angle points straight up along the positive y-axis.

Step 4 - Plot the Point

To plot the point \(3, \frac{\pi}{2}\), move 3 units from the origin along the angle \(90^{\circ}\). The point will lie on the positive y-axis, 3 units above the origin.

Key Concepts

Radial Distance

Polar coordinates use 'radial distance' to describe how far a point is from the origin. In the given example, \((3, \frac{\pi}{2})\), the radial distance is represented by \(r = 3\). This means the point is 3 units away from the origin. Think of radial distance as the length of a line from the center of a circle to any point on the edge.

• Always make sure to measure the distance accurately.
• If \(r < 0\), the point is in the opposite direction.
• Radial distance determines 'how far,' not 'where.'

Remember, the radial distance can be any non-negative number that shows the length from the origin to the point.

Angle Measurement

In polar coordinates, the angle \( \theta \) plays a crucial role. The angle is measured from the positive x-axis and describes the direction of the point. For the point \( (3, \frac{\pi}{2}) \), \( \theta = \frac{\pi}{2} \) radians.

The angle informs us about orientation:

• Angles between 0 and \( \pi \) are in quadrants I and II.
• Angles between \( \pi \) and \( 2\pi \) are in quadrants III and IV.
• Just like a clock, angles determine the 'how far around' direction.

Make sure to identify the correct quadrant when plotting points.

Radian to Degree Conversion

Angles in polar coordinates are often given in radians. It helps to convert radians to degrees for easier visualization. Use the conversion formula: \[\theta (degrees) = \theta (radians) \times \frac{180}{\pi}\] For \( \frac{\pi}{2} \) radians:

• Multiply by \( \frac{180}{\pi} \)
• \( \frac{\pi}{2} \times \frac{180}{\pi} = 90^{\circ} \)
• So, \( \frac{\pi}{2} = 90^{\circ}\)

Converting gives a clear, visual interpretation and helps place the angle confidently in the coordinate system.

Coordinate System

The polar coordinate system is different from the Cartesian system. Instead of \(x, y\) coordinates, we use (r, \( \theta \)) to locate a point. The given point \( (3, \frac{\pi}{2}) \) can be plotted by:

• Moving 3 units from the origin.
• Along an angle of \( \frac{\pi}{2} = 90^{\circ} \).
• This places the point on the y-axis, 3 units above the origin.

In this system, radial distance and angle together define the point's position. It simplifies plotting circular and rotational problems, making it a preferred choice in many applications.