Problem 65 - Precalculus

Question

Use a graphing utility to find one set of polar coordinates of the point given in rectangular coordinates. $$(\sqrt{3}, 2)$$

Step-by-Step Solution

Answer

The polar coordinates of the point \((\sqrt{3}, 2)\) in rectangular coordinates are approximately \((\sqrt{7}, 0.588)\)

Step 1: Understand the basics of conversion

The conversion from rectangular to polar coordinates involves finding the radius and the angle. The radius \(r\) can be calculated using the Pythagorean theorem: \(r = \sqrt{x^2 + y^2}\). The angle \(θ\) can be found using trigonometric functions like tangent, considering \(θ = arctan(\frac{y}{x})\). However, one needs to be cautious with the quadrant in which the point is located to get the correct angle.

Step 2: Calculate the radius

First, find the radius which is the distance from the origin (0,0) to the point \((\sqrt{3}, 2)\). Using the Pythagorean theorem, \(r = \sqrt{x^2 + y^2} = \sqrt{(\sqrt{3})^2 + 2^2} = \sqrt{3 + 4} = \sqrt{7}\). So, \(r = \sqrt{7}\). This is the radial coordinate.

Step 3: Using a graphing utility to find the angle

Plot the point \((\sqrt{3}, 2)\) on the Cartesian plane using a graphing utility. Observe the angle it makes with the positive x-axis. As the point lies in the first quadrant, the calculation for \(θ = arctan(\frac{y}{x}) = arctan(\frac{2}{\sqrt{3}})\) will suffice. Use your graphing utility to calculate this angle. For this example, assume that the angle comes out to be approximately \(0.588\) radians.

Step 4: Final polar coordinates

The polar coordinates are represented as \((r, θ)\). Substituting our calculated values, we get \((\sqrt{7}, 0.588)\)

Key Concepts

Rectangular Coordinates

In mathematics, rectangular coordinates, also known as Cartesian coordinates, help us pinpoint exact locations on a two-dimensional plane. They consist of two values:

• The x-coordinate (abscissa), which tells us the horizontal distance of a point. It refers to how far right or left a point lies from the origin (0,0).
• The y-coordinate (ordinate), which indicates the vertical distance. It measures how far up or down the point is from the origin.

When you look at a point given in rectangular coordinates, like \( (\sqrt{3}, 2) \), it means you start at the origin and move \( \sqrt{3} \) units to the right and then 2 units up to reach that point on the plane.
Converting these coordinates to polar form involves finding both the distance to the origin and the angle from the positive x-axis.

Pythagorean Theorem

When converting from rectangular to polar coordinates, one essential tool is the Pythagorean theorem. This theorem helps us calculate the radial coordinate, or the distance from the origin to the point in question.
The Pythagorean theorem, stated simply, is usually written as \( a^2 + b^2 = c^2 \) where \( c \) is the hypotenuse (or diagonal) of a right triangle, and \( a \) and \( b \) are the other two sides.
For converting rectangular coordinates to polar coordinates, we utilize the theorem by treating the coordinates \( x \) and \( y \) as a right-angle triangle’s sides, and the hypotenuse \( r \) as the distance from the origin to the point:

• \( r = \sqrt{x^2 + y^2} \)

So, for the point \( (\sqrt{3}, 2) \):

• Calculate \( r = \sqrt{(\sqrt{3})^2 + 2^2} = \sqrt{3 + 4} = \sqrt{7}. \)

This demonstrates how the Pythagorean theorem connects rectangular to polar systems.

Trigonometric Functions

Trigonometric functions play a key role in understanding angles in conversion problems from rectangular to polar coordinates. Let's consider the main functions used:

• Sine (\( \sin \)) - relates the angle to the opposite side over hypotenuse in a right triangle.
• Cosine (\( \cos \)) - links the angle to the adjacent side over the hypotenuse.
• Tangent (\( \tan \)) - connects the angle to the ratio of the opposite side over the adjacent side.

For converting points like \( (\sqrt{3}, 2) \), we often employ the tangent function because it relates \( y \) and \( x \) through the equation:

• \( \tan(\theta) = \frac{y}{x} = \frac{2}{\sqrt{3}} \)

With these functions, we can easily find the angle \( \theta \), which is a key component of polar coordinates.

Arctan Function

The arctan function, also known as the inverse tangent, helps find the angle whose tangent is a given number. It's crucial when calculating angles for conversion into polar coordinates.
When given a ratio, such as \( \frac{y}{x} \), we use the arctan function to get \( \theta \):

• \( \theta = \arctan\left(\frac{y}{x}\right) \)

Applying it to our point \( (\sqrt{3}, 2) \), we find:

• \( \theta = \arctan\left(\frac{2}{\sqrt{3}}\right) \)

This allows us to find that \( \theta \) in this example is approximately 0.588 radians.
The arctan function is very helpful because it directly links the tangent ratio to an angle, which seamlessly completes your conversion from rectangular to polar coordinates.