Problem 46 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. (There are many correct answers.) $$\left(-\frac{7}{4},-\frac{3}{2}\right)$$

Step-by-Step Solution

Answer

The point \(\left(-\frac{7}{4},-\frac{3}{2}\right)\) in polar coordinates is \(\left(r, \theta\right)\) where \(r\) and \(\theta\) are calculated in Steps 1 and 2.

Step 1: Calculating the radial distance

Calculate the radial distance \(r\) using the formula \(r = \sqrt{x^2 + y^2}\). Therefore, \(r = \sqrt{\left(-\frac{7}{4}\right)^2 + \left(-\frac{3}{2}\right)^2}\).

Step 2: Calculating the angle in radians

Calculate the angle \(\theta\) using the formula \(\theta = \arctan \left(\frac{y}{x}\right) + \pi\) since the point lies in the third quadrant. Therefore, \(\theta = \arctan \left(\frac{-\frac{3}{2}}{-\frac{7}{4}}\right) + \pi\). Remember to find the output angle in the interval \([0, 2\pi)\).

Step 3: Finding polar coordinates

The point's polar coordinates, \(\left(r, \theta\right)\), are calculated from the previous steps. These coordinates represent the same point in the polar coordinate system.

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates are used to pinpoint a location on a plane by expressing its horizontal (x) and vertical (y) values. These values are based on a grid system resembling a graph with x- and y-axes intersecting at the origin (0, 0).

In the given problem, the rectangular coordinates are \(-\frac{7}{4}, -\frac{3}{2}\). This means the point is -1.75 units along the x-axis and -1.5 units along the y-axis.

• The negative values indicate the point is in the third quadrant, where x and y are both negative.
• Understanding this system is crucial for converting to polar coordinates which describe the same location differently.

Calculating Radial Distance

In polar coordinates, the radial distance (r) is the distance from the origin to the point. To find this distance we use the Pythagorean theorem adapted to two dimensions:

\[ r = \sqrt{x^2 + y^2} \]

Plug in the given rectangular coordinates \((-\frac{7}{4}, -\frac{3}{2})\):

• First, square each coordinate:
  • \((-\frac{7}{4})^2 = \frac{49}{16}\)
  • \((-\frac{3}{2})^2 = \frac{9}{4}\)
• Add them together:
  • \(\frac{49}{16} + \frac{36}{16} = \frac{85}{16}\)
• Take the square root:
  • \(r = \sqrt{\frac{85}{16}} \approx 2.3\)

Angle Calculation in Polar Coordinates

To define a point in polar coordinates, you need an angle \(\theta\) from the positive x-axis, along with the radial distance. For points in the third quadrant, the formula for \(\theta\) is adjusted to:

\[ \theta = \arctan\left(\frac{y}{x}\right) + \pi \]

This adjustment accounts for the negative values of both x and y, positioning \(\theta\) correctly.

• Calculate \(\arctan\left(\frac{-\frac{3}{2}}{-\frac{7}{4}}\right)\)
• Resolve it to radians and then add \(\pi\) to ensure the angle lies in the correct quadrant.
• Make sure the angle stays within \([0, 2\pi)\).

Identifying the Third Quadrant

The third quadrant in a coordinate system is where both x and y values are negative. This affects both the signs and calculations you're working with when converting between rectangular and polar coordinates.

• In rectangular coordinates, moving from right to left along the x-axis is negative.
• Similarly, moving down along the y-axis is also negative.
• This is why points like \((-\frac{7}{4}, -\frac{3}{2})\) are situated in this quadrant and need specific handling during transformation to polar form.

Identifying which quadrant your point is in helps determine correct sign adjustments and angle calculations, ensuring an accurate conversion.