Problem 33 - Algebra and Trigonometry

Question

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi\). $$ (-1,1) $$

Step-by-Step Solution

Answer

Polar coordinates are \((\sqrt{2}, \frac{3\pi}{4})\).

Step 1: Understand Rectangular Coordinates

Rectangular coordinates are given as an ordered pair \((-1, 1)\). This means our x-coordinate is \(-1\) and our y-coordinate is \(1\).

Step 2: Calculate the Radius

To convert these coordinates to polar form, we need the radius \(r\), calculated using the formula \(r = \sqrt{x^2 + y^2}\). Substitute \(x=-1\) and \(y=1\) to get \(r=\sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}\).

Step 3: Determine the Angle

The angle \(\theta\) is determined using the arctangent function: \(\theta=\tan^{-1}\left(\frac{y}{x}\right)\).Substitute \(x=-1\) and \(y=1\) to calculate \(\theta = \tan^{-1}\left(\frac{1}{-1}\right) = \tan^{-1}(-1)\).

Step 4: Adjust the Angle for Correct Quadrant

The point \((-1, 1)\) is in the second quadrant, where \(\theta\) must be \(\pi/2 < \theta < \pi\). Thus, \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Step 5: Write the Polar Coordinates

Combine the results for \(r\) and \(\theta\) to write the polar coordinates. Thus, the polar coordinates are \(\left(\sqrt{2}, \frac{3\pi}{4}\right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way to locate a point on a plane by using two values: the x-coordinate and the y-coordinate. These two values form an ordered pair. In the case of the original problem, the rectangular coordinates were

• x-coordinate: -1
• y-coordinate: 1

These coordinates are crucial as they give us the exact position on the 2D plane.

The x-coordinate tells us how far to move horizontally from the origin, mainly positive to the right and negative to the left.
Simultaneously, the y-coordinate indicates how far to move vertically, positive upwards and negative downwards.

Radius Calculation

The radius in polar coordinates represents the distance from the origin to the point in question. Calculating this is like finding the hypotenuse of a right triangle, where the x and y coordinates are the other two sides.

To get the radius, we use the formula:\[ r = \sqrt{x^2 + y^2} \]In our specific example, substituting x and y with the given values (-1 and 1) leads to this calculation:

• Substitute \(-1\) for x and \(1\) for y
• Calculate \((-1)^2 = 1\)
• Calculate \((1)^2 = 1\)
• Add \(1 + 1 = 2\)
• Take the square root to find \(\sqrt{2}\)

Now, we know the radius \(r = \sqrt{2}\), which is a crucial step in transitioning to polar coordinates.

Angle Determination

Determining the angle, or \(\theta\), involves finding the direction using trigonometry. The common approach is using the arctangent function to calculate this angle, which shows the ratio between the y and x coordinates:

\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]For our pair \((-1, 1)\), we input these values as follows:

• Substitute \(1\) for y and \(-1\) for x
• Calculate the ratio \(\frac{1}{-1} = -1\)
• Apply the arctan function: \(\theta = \tan^{-1}(-1)\)

This gives an initial angle which then needs adjustment based on the quadrant where the point lies.

Quadrant Adjustments

Quadrant adjustments are necessary to ensure the angle correctly represents its actual direction on the coordinate plane. Each quadrant has specific rules for angles:

• First Quadrant: \(0 < \theta < \frac{\pi}{2}\)
• Second Quadrant: \(\frac{\pi}{2} < \theta < \pi\)
• Third Quadrant: \(\pi < \theta < \frac{3\pi}{2}\)
• Fourth Quadrant: \(\frac{3\pi}{2} < \theta < 2\pi\)

Because the point \((-1, 1)\) lies in the second quadrant, the angle calculated using \(\tan^{-1}(-1)\) initially points to the wrong direction. To correct this, adjust the angle by:

\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]Now, \(\theta\) accurately reflects the point's position, ensuring its conversion to polar coordinates is accurate and precise.