Problem 39 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (7,-7) $$

Step-by-Step Solution

Answer

The polar coordinates are \( \left( 7\sqrt{2}, \frac{7\pi}{4} \right) \).

Step 1: Identify x and y

The given rectangular coordinates are \((7, -7)\) where \(x = 7\) and \(y = -7\).

Step 2: Calculate r

Use the formula for the radius in polar coordinates: \[ r = \sqrt{x^2 + y^2} \]Substitute \(x = 7\) and \(y = -7\): \[ r = \sqrt{7^2 + (-7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

Step 3: Calculate theta

Use the formula for the angle \(\theta\):\[ \theta = \arctan\left( \frac{y}{x} \right) \]Substitute \(x = 7\) and \(y = -7\):\[ \theta = \arctan\left( \frac{-7}{7} \right) = \arctan(-1) \]Since the point is in the fourth quadrant (positive \(x\), negative \(y\)), adjust \(\theta\) to be \[ \theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \]

Step 4: Compile polar coordinates

Having computed \(r = 7\sqrt{2}\) and \(\theta = \frac{7\pi}{4}\), the polar coordinates are \( \left( 7\sqrt{2}, \frac{7\pi}{4} \right) \).

Key Concepts

Rectangular to Polar Conversion

Converting from rectangular coordinates to polar coordinates is a way of expressing a point in terms of its distance from the origin and the angle from the positive x-axis. In rectangular coordinates, a point is represented as \((x, y)\), denoting its horizontal and vertical distances from the origin respectively. In polar coordinates, a point is expressed as \((r, \theta)\), where:

• \(r\) is the radius or the distance from the origin to the point
• \(\theta\) is the angle formed with the positive x-axis

Both representations provide unique ways to describe locations in a plane. The conversion from rectangular to polar is essential when working on problems involving circular motion, physics, and engineering. Understanding this switch between coordinate systems helps to visualize different mathematical problems in a more flexible manner.
To convert coordinates, you calculate the radius and angle based on the formulas for polar coordinates, as seen in the given exercise.

Radius Calculation

The radius \(r\) in polar coordinates indicates how far the point is from the origin. If you imagine a line from the origin \((0,0)\) to the point \((x, y)\), the radius is the length of that line segment.The formula used to find the radius is:\[ r = \sqrt{x^2 + y^2} \]
So, for the point \((7, -7)\):

• First square each component: \(7^2 = 49\) and \((-7)^2 = 49\)
• Next, sum these values: \(49 + 49 = 98\)
• Finally, take the square root: \(\sqrt{98} = 7\sqrt{2}\)

The radius calculation provides a clear understanding of the point’s distance from the origin regardless of its angle from the positive x-axis.

Angle Calculation

The angle \(\theta\) in polar coordinates signifies the direction from the positive x-axis to the point. Calculating this angle can sometimes be a bit tricky, especially when negative values or specific quadrants are involved.
The basic formula for calculating the angle is:\[ \theta = \arctan\left( \frac{y}{x} \right) \]
Using this formula for the point \((7, -7)\):

• Calculate the division: \(\frac{-7}{7} = -1\)
• Determine \(\theta\): \(\arctan(-1) = -\frac{\pi}{4}\)
• Since the point resides in the fourth quadrant (positive x, negative y), adjust \(\theta\) from negative to standard form. It becomes: \(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)

This step rectifies the angle to ensure it is within the range \(0 \leq \theta < 2\pi\), which is crucial for standard polar coordinate representation. Understanding adjustments based on the quadrant helps maintain consistency when plotting the exact position in the plane.