Problem 31 - Precalculus with Calculus Previews

Question

Find polar coordinates that satisfy (a) \(r>0,-\pi<\theta \leq \pi\) (b) \(r<0,-\pi<\theta \leq \pi\) for each point with the given rectangular coordinates. $$ (-3,4) $$

Step-by-Step Solution

Answer

(5, 2.2143) for \( r > 0 \) and (-5, -0.9273) for \( r < 0 \).

Step 1: Understand the Given Rectangular Coordinates

The given rectangular coordinates are (-3, 4). These coordinates are in the form of (x, y), meaning x = -3 and y = 4. Our task is to convert these coordinates into polar coordinates in two ways, such that we satisfy the conditions given for r and θ.

Step 2: Convert Rectangular Coordinates to Polar Coordinates

Polar coordinates are given by (r, θ), where r is the distance from the origin to the point and θ (theta) is the angle from the positive x-axis.First, calculate the radius r using the formula:\[ r = \sqrt{x^2 + y^2} \]Substitute x = -3 and y = 4:\[ r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]The angle θ can be calculated using the inverse tangent function:\[ \theta = \arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{4}{-3}\right) \]Use a calculator to find θ:\[ \theta \approx 2.2143 - \pi \approx -0.9273 \] or equivalently, using radian adjustments:\[ \theta = \arctan\left(\frac{4}{-3}\right) + \pi \approx 2.2143 \]

Step 3: Find Polar Coordinates with r>0

For the case where \( r > 0 \), use the calculated values:\( r = 5 \) and \( \theta \approx 2.2143 \).Thus, the polar coordinates when \( r > 0 \) are (5, 2.2143). This satisfies the condition \( -\pi < \theta \leq \pi \).

Step 4: Find Polar Coordinates with r<0

For this condition, the radius is negative, which means the polar coordinates are represented differently but will still indicate the same point.Recalculation is necessary by converting the angle as \( \theta \approx 2.2143 - \pi \approx -0.9273 \).Now, to find coordinates with \( r < 0 \), let \( r = -5 \) and \( \theta = -0.9273 \).The polar coordinates with \( r < 0 \) are (-5, -0.9273). This also satisfies \( -\pi < \theta \leq \pi \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates form the backbone of many mathematical concepts. They consist of ordered pairs (x, y) that describe a point in a two-dimensional plane. In the exercise, we started with the rectangular coordinates (-3, 4). Here, x represents -3, and y represents 4.

• X-coordinate: The first number in the pair, indicating how far left or right the point is from the origin.
• Y-coordinate: The second number, showing how far up or down the point is from the origin.

To convert these to polar coordinates, we use these x and y values. Understanding these basics is crucial for working with various coordinate systems in math.

Inverse Tangent Function

The inverse tangent function, often expressed as \( heta = \arctan\left(\frac{y}{x}\right)\), is used to find the angle \( \theta \) in polar coordinates. It helps in turning rectangular coordinates into polar form by identifying the direction of the point from the positive x-axis.

• This involves calculating the ratio \( \frac{y}{x} \), which for our example becomes \( \frac{4}{-3} \).
• Using a calculator, \( \arctan(\frac{4}{-3}) \) gives us an initial angle, which can be adjusted depending on the quadrant in which the point resides.

The inverse tangent can return values outside the primary range \( -\pi < \theta \leq \pi \). Adjustments like adding or subtracting \( \pi \) are used to ensure \( \theta \) fits within this required interval.

Angle Conversion

Angle conversion involves modifying the angle to fit specific conditions, especially when dealing with polar coordinates. It's important to ensure that the angle \( \theta \) respects the range specified by \(-\pi < \theta \leq \pi\).

• When \( \theta \) obtained by \( \arctan \) is outside the acceptable range, adding \( \pi \) adjusts it correctly for different quadrants.
• In our solution, the angle was initially calculated as approximately 2.2143, but needed adjustment to verify it aligns with the correct quadrant.

Understanding how angles move in a circle and convert to negative or positive values help in the proper representation of coordinates, especially when switching from rectangular to polar systems.

Distance Formula

The distance formula is crucial for finding the radial component \( r \) in polar coordinates.
It finds how far a point is from the origin when starting from rectangular coordinates \((x, y)\). Using the formula \( r = \sqrt{x^2 + y^2} \), we calculate \( r \). In this example:

• Substituting values \( x = -3 \) and \( y = 4 \) into the formula gives \( r = \sqrt{(-3)^2 + 4^2} = 5 \).

This value of \( r \) signifies the straight-line distance from the origin to the point \((-3, 4)\). Though straightforward, this powerful formula is a fundamental building block of coordinate conversion, ensuring the calculation is efficient and accurate for the transformation process.