Problem 39 - 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 \(\mathbf{0} \leq \boldsymbol{\theta}<\mathbf{2} \pi\) $$(-3,4)$$

Step-by-Step Solution

Answer

The two sets of polar coordinates for the point (-3, 4) are (5, \(\arctan(-\frac{4}{3}) + \pi)\) and (5, \(\arctan(-\frac{4}{3}) + \pi + 2\pi\))

Step 1: Convert rectangular coordinates to polar coordinates

First convert the rectangular coordinate (-3, 4) to polar coordinates using the following formulas: \( r = \sqrt{(-3)^2 + 4^2} \) to find r, and \( \theta = arctan(\frac{y}{x}) = arctan(\frac{4}{-3}) \) to find \(\theta\). Designate \(\theta_1\) as the principal angle.

Step 2: Calculate r

By putting the values of x and y in the formula, \( r = \sqrt{9 + 16} = \sqrt{25} = 5 \). So, the r-coordinate is 5.

Step 3: Calculate First \(\theta_1\)

Substituting y=4 and x=-3 into the formula: \( \theta_1 = arctan(-\frac{4}{3}) \). This will generate an angle in the 4th quadrant, but because we are in the 2nd quadrant (since x is negative and y is positive), we add \(\pi\) to get the correct angle. So, \( \theta_1 = arctan(-\frac{4}{3}) + \pi \).

Step 4: Identify another valid set of polar coordinates

Recall that polar coordinates are not unique. The second set of polar coordinates could be found by adding \(2\pi\) to the original angle: \( \theta_2 = \theta_1 + 2\pi \).

Step 5: Calculate Second \( \theta_2 \)

Substituting \(\theta_1\) into the formula, we get: \( \theta_2 = arctan(-\frac{4}{3}) + \pi + 2\pi \).

Key Concepts

Rectangular to Polar Conversion

Converting rectangular coordinates to polar coordinates involves finding the distance from the origin to the point (denoted as r) and the angle between the positive x-axis and the line connecting the origin to the point (denoted as theta, \theta).

In our example with the coordinate (-3, 4), we calculate r using the Pythagorean theorem, which in a polar context means squaring both the x and y rectangular coordinates, summing them, and then taking the square root: \( r = \sqrt{x^2 + y^2} \).

In our exercise, this comes out to be:\( r = \sqrt{(-3)^2 + (4)^2} = \sqrt{25} = 5 \).

To find \(\theta\), we take the arctan of y/x, or in this case, \(\theta = \arctan\left(\frac{y}{x}\right)\). As \(\arctan\) of a negative ratio gives us an angle in the fourth quadrant and we know the point is actually in the second quadrant (since x is negative and y is positive), we adjust by adding \(\pi\) to the base angle to properly orient the angle relative to the x-axis.

In practical terms, the original angle would be \( \theta_1 = \arctan\left(\frac{-4}{3}\right) + \pi \). This conversion is crucial as it allows for the description of the same point in another coordinate system widely used in fields like physics and engineering for its convenience in dealing with rotated frames, circular, or spiral paths.

Polar Coordinate System

The polar coordinate system is a two-dimensional coordinate system in which each point on a plane is identified by a distance from a reference point and an angle from a reference direction. The reference point is known as the pole, akin to the origin in rectangular coordinates.

Each point is represented as \( (r, \theta) \) in the polar coordinate system, where \( r \) is the radial coordinate - the distance from the pole, and \( \theta \) is the angular coordinate – the angle related to the positive x-axis.

One of the remarkable features of this system is the multiplicity of coordinates for a single point; for example, adding \( 2\pi \) radians to an angle gives you the same direction since \( 2\pi \) radians equate to a full circle. This is evident from our exercise, where the point given by the second set of polar coordinates is obtained by adding \( 2\pi \) to \( \theta_1 \), resulting in \( \theta_2 = \theta_1 + 2\pi \). This emphasizes the periodic nature of the polar coordinate system where, similar to how a 360° rotation leads you in the same direction, adding \( 2\pi \) radians to the angle leads to equivalent positioning.

Radian Measure

The radian measure is a way of expressing angles, different from the more commonly known degree measure. One radian is the angle created when the radius is wrapped around the circumference of a circle, thus defining a segment equal in length to the radius itself. It's the SI unit for measuring angles.

In mathematical terms, there are \( 2\pi \) radians in a full circle, corresponding to 360 degrees. To convert from degrees to radians, we multiply by \( \frac{\pi}{180} \) and vice-versa.

Understanding radians is crucial in our topic because, in polar coordinates, the angle \( \theta \) is typically expressed in radians. This allows for a more natural integration of trigonometric functions with calculus and provides a direct correlation between angular and linear measurements. As observed in the exercise, the calculated angle \( \theta \) needs to be in radians to use the function arctan effectively, and to respect the periodicity inherent to angular measures in trigonometry.