Problem 9 - Algebra and Trigonometry

Question

Plot the point that has the given polar coordinates. Then give two other polar coordinate representations of the point, one with \(r<0\) and the other with \(r>0\) . $$ (3, \pi / 2) $$

Step-by-Step Solution

Answer

Other coordinates are \((-3, 3\pi/2)\) and \((3, 5\pi/2)\).

Step 1: Understanding Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle in radians measured counterclockwise from the positive x-axis.

Step 2: Plot the Initial Point

The point \((3, \pi/2)\) means we need to move 3 units away from the origin in the direction of the angle \(\pi/2\), which lies on the positive y-axis. Therefore, this point is located directly above the origin at 3 units on the positive y-axis.

Step 3: Finding Another Coordinate with r<0

For \(r < 0\), we use \((r, \theta + \pi)\). Here, \(r = -3\) and the angle \(\theta = \pi/2 + \pi = 3\pi/2\). This moves the point to the opposite side, which places it at the same position as the original point.

Step 4: Finding Another Coordinate with r>0

For \(r > 0\), we can use \(r = 3\) and find another equivalent angle by adding \(2\pi\) to the angle: \(\theta = \pi/2 + 2\pi = 5\pi/2\). This keeps the point at the same location.

Key Concepts

Angle in Radians

In polar coordinates, the angle is typically given in radians, which is a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians focus on the relationship between the radius and the arc of a circle.- One radian is the angle formed when the length of the arc is equal to the radius.- This makes a full circle equal to \(2\pi\) radians or approximately 6.283 radians.- Therefore, \( \pi/2 \) radians is a quarter of a circle or 90 degrees.Understanding radians helps in connecting different forms of rotation and circular movements, which is essential when working with polar coordinates. In the given exercise, the angle \( \pi/2 \) is particularly important as it points directly upwards in a Cartesian plane.

Plotting Points

When plotting points using polar coordinates, remember that each point is defined by two values: the radius \(r\) and the angle \(\theta\). The radius tells us how far to go from the origin, and the angle tells us in which direction to go. - Start at the origin and face along the positive x-axis.- Rotate counterclockwise by the angle \(\theta\).- Move outwards by the radius \(r\).In our example, the polar coordinates \((3, \pi/2)\) mean:- The point is 3 units away from the origin.- The angle \( \pi/2 \) points the location straight up, along the positive y-axis.Once you master these steps, converting between polar and Cartesian coordinates becomes much easier.

Coordinate Transformation

Coordinate transformation is the process of converting points between different coordinate systems, such as from polar to Cartesian coordinates or vice versa.

• Polar to Cartesian: Use the formulas \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• Cartesian to Polar: Use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \arctan\left(\frac{y}{x}\right) \).

In our exercise:- The original point in polar coordinates \((3, \pi/2)\) transforms to Cartesian coordinates by calculating: \[ x = 3 \cos(\pi/2) = 0 \] \[ y = 3 \sin(\pi/2) = 3 \]This transformation verifies the location at \((0, 3)\) on the Cartesian plane. By understanding these transformation rules, we can easily shift between systems, ensuring a wide range of applications in mathematics and engineering.