Problem 56 - Precalculus

Question

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-1, \sqrt{3})$$

Step-by-Step Solution

Answer

The polar coordinates of the point (-1, \(\sqrt{3}\)) in the rectangular coordinate system are (2, 120°)

Step 1: Computation of r

Find the magnitude (r) from rectangular coordinates using the formula \(r = \sqrt{x^2 + y^2}\). Provided coordinates are: x = -1 and y = \(\sqrt{3}\), the magnitude can be calculated as \(r = \sqrt{(-1)^2 + (\sqrt{3})^2} = 2\)

Step 2: Computing θ

Calculate the angle using the formula \(θ = atan2(y, x)\). Here, x = -1 and y = \(\sqrt{3}\). Therefore, \(θ = atan2(\sqrt{3}, -1)\). Since \(atan2(\sqrt{3},-1)\) returns an angle in the third quadrant, we must add 180° to the result to put the angle in the correct place, in quadrant II. Hence, \(θ = 180° + atan2(\sqrt{3}, -1)\). This gives \(θ = 120°\)

Step 3: Write the Coordinates in Polar Form

The final step is to write the polar coordinates using the (r, θ) format. Substituting the values from steps 1 and 2, the polar coordinates of the point (-1, \(\sqrt{3}\)) are thus (2, 120°)

Key Concepts

Rectangular Coordinates

Rectangular coordinates are often referred to as Cartesian coordinates. These coordinates are used to define the position of a point in two-dimensional space using two perpendicular axes, usually labeled as the x-axis and y-axis.
The rectangular coordinates are written as \((x, y)\), where:

• The x-coordinate represents the horizontal distance from the origin along the x-axis.
• The y-coordinate represents the vertical distance from the origin along the y-axis.

For example, the point \((-1, \sqrt{3})\) specifies a position in the coordinate plane. Here, \(-1\) is the coordinate on the x-axis, indicating that the point moves 1 unit to the left from the origin, and \(\sqrt{3}\) is the coordinate on the y-axis, meaning the point moves up by \(\sqrt{3}\) units.

Coordinate Conversion

Converting between coordinate systems is a common task in mathematics, especially when dealing with curves and rotational systems. In this specific exercise, we are converting from rectangular to polar coordinates.
Polar coordinates define a point in terms of its distance from the origin, \(r\), and the angle \(θ\) it forms with the positive x-axis.
To perform the conversion:

• Calculate \(r\), the magnitude, using the formula \(r = \sqrt{x^2 + y^2}\). This formula comes from the Pythagorean Theorem.
• Determine \(θ\), the angle, using \(θ = \text{atan2}(y, x)\). This function is beneficial as it accounts for the quadrant in which the point is located.

Considering \((-1, \sqrt{3})\), we first find \(r = 2\). Then, using \(\text{atan2}(\sqrt{3}, -1)\) yields an angle in the third quadrant, which requires adding 180° to adjust for the actual position.
This conversion approach is crucial for problems involving angles and curves, where polar coordinates provide a more intuitive understanding.

Trigonometry

Trigonometry plays an essential role in coordinate conversion, as it helps us understand the relationships between angles and coordinates.
The tangent function, particularly \(\text{atan2}(y, x)\), is key in the conversion process, determining the angle \(θ\) between the radius vector and the positive x-axis.
Understanding trigonometric functions allows us to:

• Find angles when given side lengths, which is the case in the conversion from rectangular to polar coordinates.
• Use inverse-tangent, \(\tan^{-1}\), specifically \(\text{atan2}\) for its quadrant-correcting properties, to ensure the angle reflects the correct direction from the x-axis as seen in the problem.

In the conversion, the angle \(θ\) was identified as \(120°\), acknowledging the angles' trigonometric rules, critical in correctly representing the direction of the line from the origin to point \((x, y)\).