Problem 6 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

(a) (1, 1); (b) (1, 0); (c) (0, 0); (d) (-1, -1); (e) (3√3/2, -3/2); (f) (3, 4); (g) (1, 0); (h) (-√3, 3).

Step 1: Understand Polar to Cartesian Conversion

Polar coordinates are given in the form \((r, \theta)\). To convert polar coordinates to Cartesian coordinates \((x, y)\), we use the formulas: \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

Step 2: Convert Point (a)

For the point \((\sqrt{2}, \pi / 4)\):1. Calculate \(x = \sqrt{2} \cos(\pi / 4) = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1\).2. Calculate \(y = \sqrt{2} \sin(\pi / 4) = \sqrt{2} \times \frac{\sqrt{2}}{2} = 1\).Thus, the Cartesian coordinates are \((1, 1)\).

Step 3: Convert Point (b)

For the point \((1, 0)\):1. Calculate \(x = 1 \cos(0) = 1 \times 1 = 1\).2. Calculate \(y = 1 \sin(0) = 1 \times 0 = 0\).Thus, the Cartesian coordinates are \((1, 0)\).

Step 4: Convert Point (c)

For the point \((0, \pi / 2)\):1. Calculate \(x = 0 \cos(\pi / 2) = 0 \times 0 = 0\).2. Calculate \(y = 0 \sin(\pi / 2) = 0 \times 1 = 0\).Thus, the Cartesian coordinates are \((0, 0)\).

Step 5: Convert Point (d)

For the point \((-\sqrt{2}, \pi / 4)\):1. Calculate \(x = -\sqrt{2} \cos(\pi / 4) = -\sqrt{2} \times \frac{\sqrt{2}}{2} = -1\).2. Calculate \(y = -\sqrt{2} \sin(\pi / 4) = -\sqrt{2} \times \frac{\sqrt{2}}{2} = -1\).Thus, the Cartesian coordinates are \((-1, -1)\).

Step 6: Convert Point (e)

For the point \((-3, 5\pi / 6)\):1. Calculate \(x = -3 \cos(5\pi / 6) = -3 \times \left(-\frac{\sqrt{3}}{2}\right) = \frac{3\sqrt{3}}{2}\).2. Calculate \(y = -3 \sin(5\pi / 6) = -3 \times \frac{1}{2} = -\frac{3}{2}\).Thus, the Cartesian coordinates are \(\left(\frac{3\sqrt{3}}{2}, -\frac{3}{2}\right)\).

Step 7: Convert Point (f)

For the point \((5, \tan^{-1}(4/3))\):1. Calculate \(x = 5 \cos(\tan^{-1}(4/3)) = 5 \times \frac{3}{5} = 3\). (Using cosine identity of angle with given tangent)2. Calculate \(y = 5 \sin(\tan^{-1}(4/3)) = 5 \times \frac{4}{5} = 4\). (Using sine identity of angle with given tangent)Thus, the Cartesian coordinates are \((3, 4)\).

Step 8: Convert Point (g)

For the point \((-1, 7\pi)\):1. Since \(\cos(7\pi) = -1\) and \(\sin(7\pi) = 0\) (since \(7\pi\) corresponds to an odd multiple of \(\pi\)),Calculate \(x = -1 \times -1 = 1\).Calculate \(y = -1 \times 0 = 0\).Thus, the Cartesian coordinates are \((1, 0)\).

Step 9: Convert Point (h)

For the point \((2\sqrt{3}, 2\pi / 3)\):1. Calculate \(x = 2\sqrt{3} \cos(2\pi / 3) = 2\sqrt{3} \times \left(-\frac{1}{2}\right) = -\sqrt{3}\).2. Calculate \(y = 2\sqrt{3} \sin(2\pi / 3) = 2\sqrt{3} \times \frac{\sqrt{3}}{2} = 3\).Thus, the Cartesian coordinates are \((-\sqrt{3}, 3)\).

Key Concepts

Trigonometry Basics

Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles, especially right triangles. This area is fundamental for converting between polar and Cartesian coordinates. When dealing with right triangles, we use three primary functions: sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions are defined as ratios of the sides of the triangle:

• Sine is the ratio of the opposite side to the hypotenuse.
• Cosine is the ratio of the adjacent side to the hypotenuse.
• Tangent is the ratio of the opposite side to the adjacent side.

Understanding these functions is crucial when dealing with angles and can vastly help in visualizing how polar coordinates relate to Cartesian ones. By knowing an angle and one side in a triangle, trigonometric functions allow us to calculate the lengths of the other sides, making explicit the connection between angle measures and rectangular coordinates.

Polar Coordinates Fundamentals

Polar coordinates offer a way to express a point in a plane using a distance and an angle, often denoted as (\( r, \theta \)). Here, \( r \) represents the radial distance from the origin, and \( \theta \) is the angle measured from the positive x-axis. This system is particularly useful in scenarios where circular or rotational symmetry is evident like with ripples in water or things orbiting around a center.
When converting to Cartesian coordinates, it's vital to grasp that we're turning this circular system into a rectilinear grid, where each point is given by (x, y). This involves straightforward trigonometric projections—extending trigonometric understanding to determine the horizontal and vertical displacement from the origin. By identifying \( r \) as the hypotenuse of a right triangle with horizontal (\( x = r \cos(\theta) \)) and vertical (\( y = r \sin(\theta) \)) sides, we bridge polar coordinates to Cartesian coordinates, simplifying the representation of a location graphically.

Cartesian Coordinates Overview

Cartesian coordinates, named after René Descartes who pioneered this concept, describe a system using an orthogonal grid to represent points in a plane. Each point has a set of coordinates, (x, y), which specifies its unique position based on horizontal (x-axis) and vertical (y-axis) distance from the origin of the grid, where these axes intersect at (0,0).
This coordinate system is fundamental for geometry, physics, engineering, and computer graphics, showing versatility in practical applications.

• x-coordinates tell us how far left or right we move from the origin.
• y-coordinates tell us how far up or down we move from the origin.

When dealing with conversions from polar to Cartesian, we effectively align each point's polar representation to the Cartesian grid. This transfer requires using trigonometric relations, seamlessly transforming radial and angular data into understandable and tangible visuals. This conversion is key in situations ranging from physics problems to graphical displays on computers, highlighting the Cartesian system's role in connecting abstract mathematical explanations to visible, real-world frameworks.