Problem 40 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in rectangular coordinates and find two sets of polar coordinates for the point for \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(3,-1)$$

Step-by-Step Solution

Answer

The two sets of polar coordinates for the point are \((\sqrt{10}, 5.962)\) and \((\sqrt{10}, 12.244)\).

Step 1: Convert to Polar Coordinates

First, find the magnitude of the radius using the formula \(r=\sqrt{x^{2}+y^{2}}\). Here \(x=3\) and \(y=-1\), so the magnitude of the radius \(r=\sqrt{(3)^{2}+(-1)^{2}}=\sqrt{10}\). Next, find the angle \(\theta\) (in radians) using \(\theta =\tan^{-1}\frac{y}{x}\). Here, \(\theta =\tan^{-1}\frac{-1}{3}\), which equals to \(-0.321\) radians. Since \(\theta\) is negative, to get the angle in the first revolution, add \(2 \pi\) to it. That will give us \(\theta_{1}=2\pi - 0.321 = 5.962\) radians.

Step 2: Find Second Set of Polar Coordinates

We have found one set of polar coordinates \((\sqrt{10}, 5.962)\). However, since the conditions \(0 \leq \theta <2\pi\) allows for another possible set of polar coordinates for the same point, we can find another valid \(\theta\) by adding \(2 \pi\) to \(\theta_{1}\). That gives us \(\theta_{2} = 5.962 + 2\pi = 12.244\). So, the second set of polar coordinates are \((\sqrt{10}, 12.244)\).

Step 3: Plotting the Point

Finally, plot the point on the polar coordinate system. Start at the origin, move out a distance of \(\sqrt{10}\) away from the origin along the positive x-axis, then rotate by 5.962 radians (or 12.244 radians) counterclockwise. That's where the point \((3,-1)\) would lie in polar coordinates.

Key Concepts

Rectangular Coordinates

In the world of coordinate systems, rectangular coordinates are one of the most common ways to locate a point in a plane. Also known as Cartesian coordinates, they define a point using a pair of numerical values:

• The first value, known as the x-coordinate, represents the horizontal position.
• The second value, y-coordinate, represents the vertical position.

For example, in the point (3, -1), 3 is the x-coordinate, indicating how far right the point is from the point of origin (0,0). The value -1 is the y-coordinate, showing the position below the origin. By using these coordinates, you can easily find and plot points on a graph.

Coordinate Conversion

Switching from rectangular coordinates to polar coordinates involves a few straightforward calculations. Polar coordinates are expressed with a radius and an angle, represented as \(r, \theta\).

• The radius \(r\) is the distance from the origin, and it is calculated using the formula \(r = \sqrt{x^{2}+y^{2}}\).
• The angle \(\theta\) is measured in radians, calculated with \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).

By converting from rectangular to polar coordinates, you translate a point based on its distance and direction from the origin, offering a different way to understand its location.

Radians

In mathematics, angles can be measured in degrees or radians. Radians offer a more natural way of measurement, especially in the context of trigonometry and calculus. One complete circle is \(2\pi\) radians, corresponding to 360 degrees.

• An angle of \(\pi\) radians is equivalent to 180 degrees.
• Smaller angles are fractions of \(\pi\). For instance, \(\frac{\pi}{2}\) radians is 90 degrees.

When converting rectangular coordinates to polar, the angle \(\theta\) is typically expressed in radians. This conversion involves ensuring that \(\theta\) falls within the standard range of \(0\leq \theta <2\pi\) for accurate plotting and understanding.

Trigonometry

Trigonometry is a branch of mathematics that delves into the relationships between the sides and angles of triangles. It's essential when working with polar coordinates, as it involves calculating angles and distances.

• Using the tangent function, \(\tan \theta = \frac{y}{x}\), allows you to find the angle \(\theta\).
• Pythagoras' theorem helps calculate the radius \(r\) with \(r = \sqrt{x^2 + y^2}\).

Trigonometry shines by connecting angles and distances, enabling conversions between different coordinate systems and improving understanding of spatial relations.