Question

You can locate a point in a plane using Cartesian coordinates \((x, y),\) where \(|x|\) is the distance from the \(y\) -axis and \(|y|\) is the distance from the \(x\) -axis. You can also locate a point in a plane using \((r, \theta)\) where \(r, r \geq 0,\) is the distance from the origin and \(\theta\) is the angle of rotation from the positive \(x\) -axis. These are known as polar coordinates. Determine the polar coordinates for each point. a) \(\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\) b) \(\left(-\frac{\sqrt{3}}{2},-\frac{1}{3}\right)\) c) (2,2) d) (4,-3)

Short answer

a) \( (1, \frac{\pi}{4}) \) b) \( \left(\frac{\sqrt{31}}{6}, \pi + \arctan\left(\frac{2\sqrt{3}}{9}\right)\right) \) c) \( (2\sqrt{2}, \frac{\pi}{4}) \) d) \( (5, 2\pi - \arctan\left(\frac{3}{4}\right)) \)

Step-by-step solution

- Understand the Conversion

To convert from Cartesian coordinates \( (x, y) \) to polar coordinates \( (r, \theta) \), use the formulas: \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]. Ensure to adjust \( \theta \) based on the quadrant of the point.

- Convert Point (a)

For the point \( \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) \): \[ r = \sqrt{\left(\frac{\sqrt{2}}{2}\right)^2 + \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{\frac{2}{4} + \frac{2}{4}} = \sqrt{1} = 1 \] \[ \theta = \arctan\left(\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\right) = \arctan(1) = \frac{\pi}{4} \]. So, the polar coordinates are \( (1, \frac{\pi}{4}) \).

- Convert Point (b)

For the point \( \left(-\frac{\sqrt{3}}{2}, -\frac{1}{3}\right) \): \[ r = \sqrt{\left(-\frac{\sqrt{3}}{2}\right)^2 + \left(-\frac{1}{3}\right)^2} = \sqrt{\frac{3}{4} + \frac{1}{9}} = \sqrt{\frac{27}{36} + \frac{4}{36}} = \sqrt{\frac{31}{36}} = \frac{\sqrt{31}}{6} \] \[ \theta = \arctan\left(\frac{-\frac{1}{3}}{-\frac{\sqrt{3}}{2}}\right) = \arctan\left(\frac{2}{3\sqrt{3}}\right) = \arctan\left(\frac{2\sqrt{3}}{9}\right) \]. Since the point is in the third quadrant, \( \theta = \pi + \arctan\left(\frac{2\sqrt{3}}{9}\right) \). So, the polar coordinates are \( \left(\frac{\sqrt{31}}{6}, \pi + \arctan\left(\frac{2\sqrt{3}}{9}\right)\right) \).

- Convert Point (c)

For the point \( (2, 2) \): \[ r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \] \[ \theta = \arctan\left(\frac{2}{2}\right) = \arctan(1) = \frac{\pi}{4} \]. So, the polar coordinates are \( (2\sqrt{2}, \frac{\pi}{4}) \).

- Convert Point (d)

For the point \( (4, -3) \): \[ r = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \] \[ \theta = \arctan\left(\frac{-3}{4}\right) \]. Since the point is in the fourth quadrant, \( \theta = 2\pi - \arctan\left(\frac{3}{4}\right) \). So, the polar coordinates are \( (5, 2\pi - \arctan\left(\frac{3}{4}\right)) \).

Key concepts

Coordinate Conversion

Converting between Cartesian and polar coordinates can seem complicated, but with practice, it becomes straightforward. Cartesian coordinates \( (x, y) \) describe a point's position in the plane in terms of horizontal and vertical distances from the origin. On the other hand, polar coordinates \( (r, \theta) \) use the distance from the origin (r) and the angle from the positive x-axis (\theta). To convert from Cartesian to polar coordinates, use these formulas:
\[ r = \sqrt{x^2 + y^2} \] This calculates the distance (r) from the origin to the point. Meanwhile, the angle (\theta) is found with:
\[ \theta = \arctan\left(\frac{y}{x}\right) \] It's crucial to consider the quadrant where the point is located to adjust the angle correctly. Polar coordinates offer an alternative way to represent points, especially useful in scenarios involving rotations and circular paths.

Cartesian Coordinates

Cartesian coordinates are a familiar way to pinpoint a location in a plane. They are written as \( (x, y) \), where x denotes the horizontal distance from the y-axis and y represents the vertical distance from the x-axis. In other words:
\- x: Moves left or right
\- y: Moves up or down
Every point on the plane corresponds to a unique pair of values (x, y). These coordinates are particularly useful for plotting graphs and solving equations. For example, the point (2, 3) is located 2 units to the right of the y-axis and 3 units up from the x-axis.

Angle Measurement

Angles in polar coordinates are measured from the positive x-axis. This angle, denoted as \( \theta \), can be measured in degrees or radians. For polar coordinates:
\- \( r \): Length from the origin to the point
\- \( \theta \): Counterclockwise angle from the positive x-axis
Quadrants are essential in determining the correct angle:
\- **First Quadrant (0 to \ \frac{\text{π}}{2} radians)**: Both x and y are positive.
\- **Second Quadrant (\ \frac{\text{π}}{2} to \ π radians)**: x is negative and y is positive.
\- **Third Quadrant (\ π to \ \frac{3\text{π}}{2} radians)**: Both x and y are negative.
\- **Fourth Quadrant (\ \frac{3\text{π}}{2} to 2\ \text{π} radians)**: x is positive and y is negative.
Knowing the quadrant of your Cartesian coordinates helps adjust the angle correctly when converting to polar coordinates.