Problem 6 - Thomas Calculus

Question

Find the Cartesian coordinates of the following points (given in polar coordinates). $$\begin{array}{ll}{\text { a. }(\sqrt{2}, \pi / 4)} & {\text { b. }(1,0)} \\\ {\text { c. }(0, \pi / 2)} & {\text { d. }(-\sqrt{2}, \pi / 4)}\end{array}$$ $$\begin{array}{ll}{\text { e. }(-3,5 \pi / 6)} & {\text { f. }\left(5, \tan ^{-1}(4 / 3)\right)} \\ {\text { g. }(-1,7 \pi)} & {\text { h. }(2 \sqrt{3}, 2 \pi / 3)}\end{array}$$

Step-by-Step Solution

Answer

a. (1, 1), b. (1, 0), c. (0, 0), d. (-1, -1), e. (\(\frac{3\sqrt{3}}{2},-\frac{3}{2}\)), f. (3, 4), g. (1, 0), h. (-\sqrt{3}, 3).

Step 1: Understanding Polar and Cartesian Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis. Cartesian coordinates are in the form \((x, y)\) and can be derived as follows:- \(x = r \cos(\theta)\)- \(y = r \sin(\theta)\)This conversion formula will be used for each given point in polar coordinates.

Step 2: Convert Point a

Given point in polar coordinates: \((\sqrt{2}, \pi / 4)\).- \(x = \sqrt{2} \cos(\pi / 4) = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1\)- \(y = \sqrt{2} \sin(\pi / 4) = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1\)The Cartesian coordinates are \((1, 1)\).

Step 3: Convert Point b

Given point in polar coordinates: \((1, 0)\).- \(x = 1 \cos(0) = 1\)- \(y = 1 \sin(0) = 0\)The Cartesian coordinates are \((1, 0)\).

Step 4: Convert Point c

Given point in polar coordinates: \((0, \pi/2)\).- \(x = 0 \cos(\pi/2) = 0\)- \(y = 0 \sin(\pi/2) = 0\)The Cartesian coordinates are \((0, 0)\).

Step 5: Convert Point d

Given point in polar coordinates: \((-\sqrt{2}, \pi / 4)\).- \(x = -\sqrt{2} \cos(\pi / 4) = -1\)- \(y = -\sqrt{2} \sin(\pi / 4) = -1\)The Cartesian coordinates are \((-1, -1)\).

Step 6: Convert Point e

Given point in polar coordinates: \((-3, 5\pi / 6)\).- \(x = -3 \cos(5\pi / 6) = -3 \times (-\frac{\sqrt{3}}{2}) = \frac{3\sqrt{3}}{2}\)- \(y = -3 \sin(5\pi / 6) = -3 \times \frac{1}{2} = -\frac{3}{2}\)The Cartesian coordinates are \((\frac{3\sqrt{3}}{2}, -\frac{3}{2})\).

Step 7: Convert Point f

Given point in polar coordinates: \((5, \tan^{-1}(4/3))\), where \(\tan(\theta) = \frac{4}{3}\).- Cosine and sine for angle \(\theta\):- \(x = 5 \cos(\theta) = 5 \times \frac{3}{5} = 3\)- \(y = 5 \sin(\theta) = 5 \times \frac{4}{5} = 4\)The Cartesian coordinates are \((3, 4)\).

Step 8: Convert Point g

Given point in polar coordinates: \((-1, 7\pi)\).- \(x = -1 \cos(7\pi) = -1 \times (-1) = 1\)- \(y = -1 \sin(7\pi) = -1 \times 0 = 0\)The Cartesian coordinates are \((1, 0)\).

Step 9: Convert Point h

Given point in polar coordinates: \((2\sqrt{3}, 2\pi / 3)\).- \(x = 2\sqrt{3} \cos(2\pi / 3) = 2\sqrt{3} \times (-\frac{1}{2}) = -\sqrt{3}\)- \(y = 2\sqrt{3} \sin(2\pi / 3) = 2\sqrt{3} \times \frac{\sqrt{3}}{2} = 3\)The Cartesian coordinates are \((-\sqrt{3}, 3)\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in two-dimensional space using a radial distance and an angle. This system is particularly useful when dealing with circular or rotational symmetries.
In polar coordinates, each point is described by two values:

• \( r \) (radius): the distance from the origin to the point.
• \( \theta \) (theta): the angle from the positive x-axis to the point.

Because they use angles, polar coordinates are ideal for problems involving rotations or curves like circles and spirals.
An example of a polar coordinate is \((\sqrt{2}, \pi / 4)\), where \(\sqrt{2}\) is the radial distance and \(\pi / 4\) is the angle.

Cartesian Coordinates

Also known as rectangular coordinates, Cartesian coordinates provide a straightforward way to pinpoint a location in a plane using perpendicular axes. These coordinates use a grid system and are denoted by two values:

• \( x \): the horizontal distance from the origin.
• \( y \): the vertical distance from the origin.

In a Cartesian system, each point is represented as \((x, y)\). This coordinate system is useful because it simplifies calculations for lines and other linear shapes in algebra and geometry.
For example, the polar point \((1, 1)\) in Cartesian coordinates indicates a position that is 1 unit on the x-axis and 1 unit on the y-axis, beginning from the origin.

Coordinate Transformation

Coordinate transformation between polar and Cartesian systems involves using trigonometry to convert between the systems. This conversion is useful for solving problems that benefit from one representation over the other.
For converting polar \((r, \theta)\) to Cartesian \((x, y)\):

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

These formulas leverage the definitions of cosine and sine, which relate an angle to the respective sides of a right triangle.
Conversely, to convert Cartesian \((x, y)\) to polar \((r, \theta)\), use:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

These transformations are essential in fields ranging from engineering to computer graphics, where different representations can simplify problem-solving.