Problem 15 - Precalculus

Question

Convert each point to exact polar coordinates. Assume that \(0 \leq \theta<2 \pi.\) $$(-4,4)$$

Step-by-Step Solution

Answer

The polar coordinates are \((4\sqrt{2}, \frac{3\pi}{4})\)."

Step 1: Identify Cartesian coordinates

The given point is in the Cartesian coordinate system: \((-4, 4)\). Here, \(x = -4\) and \(y = 4\).

Step 2: Calculate the radial distance

The radial distance \(r\) from the origin to the point \((-4, 4)\) is calculated using the formula \(r = \sqrt{x^2 + y^2}\). Plug in \(x\) and \(y\): \[r = \sqrt{(-4)^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\]

Step 3: Determine the angle in radians

The angle \(\theta\) is calculated using the tangent formula \(\theta = \arctan\left(\frac{y}{x}\right)\), noting that adjustments may be needed based on the quadrant. Here, \[\theta = \arctan\left(\frac{4}{-4}\right) = \arctan(-1)\]The point \((-4, 4)\) is in the second quadrant, so \(\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}\).

Step 4: Write the polar coordinates

Polar coordinates are written as \((r, \theta)\). We have found: \[r = 4\sqrt{2}\] and \[\theta = \frac{3\pi}{4}\].Thus, the polar coordinates are \((4\sqrt{2}, \frac{3\pi}{4})\).

Key Concepts

Cartesian coordinate system

The Cartesian coordinate system is a two-dimensional grid defined by an x-axis and a y-axis. It's the most common method to represent points on a plane.

• The horizontal axis is called the x-axis, and the vertical axis is called the y-axis.
• Each point on the plane can be identified by a pair of numerical coordinates: (x, y).
• The coordinates indicate the point’s distance from the axes.

In a broader sense, the Cartesian coordinate system provides a straightforward way to pinpoint locations efficiently. For our problem's point,
(-4, 4), it means that from the origin, this point is 4 units to the left and 4 units up.

radial distance

In polar coordinates, the radial distance, often represented as \( r \), is the straight-line distance from the origin to the point in the plane.

• It is calculated using the formula: \( r = \sqrt{x^2 + y^2} \).
• This formula is derived from the Pythagorean theorem.
• It always equals or exceeds zero.

For the point (-4, 4), the radial distance is computed as \( r = \sqrt{(-4)^2 + 4^2} = 4\sqrt{2} \). This distance tells us how far the point is from the origin directly.

angle conversion

Converting a point into polar coordinates requires determining the angle \( \theta \) between the positive x-axis and the line segment connecting the origin to the point.

• Angle \( \theta \) can be computed using the inverse tangent function: \( \theta = \arctan\left(\frac{y}{x}\right) \).
• However, adjustments must be made depending on the location of the angle in different quadrants.
• Angles are generally measured in radians in polar coordinates, where one full revolution (360 degrees) equals \( 2\pi \) radians.

For the point (-4, 4), after calculating \( \arctan(-1) \), adjustments for the second quadrant lead to \( \theta = \frac{3\pi}{4} \).

quadrants

The Cartesian coordinate system is divided into four sections known as quadrants. Each quadrant helps determine the appropriate angle in polar coordinates.

• Quadrant I: where both \( x \) and \( y \) coordinates are positive.
• Quadrant II: where \( x \) is negative, and \( y \) is positive.
• Quadrant III: where both \( x \) and \( y \) are negative.
• Quadrant IV: where \( x \) is positive, and \( y \) is negative.

For the point in question (-4, 4), it is located in Quadrant II because \( x = -4 \) and \( y = 4 \). Adjustments to the angle calculation need to consider that the initial output from tools like \( \arctan \) typically assumes that angles are generated with respect to Quadrant I, affecting how we reach the angle of \( \frac{3\pi}{4} \).