Problem 67 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (1.3,-2.1) $$

Step-by-Step Solution

Answer

The polar coordinates are approximately (2.47, 5.26).

Step 1: Find the radius r

The radius (r) can be found using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \] Plug in the given coordinates (1.3, -2.1): \[ r = \sqrt{1.3^2 + (-2.1)^2} = \sqrt{1.69 + 4.41} = \sqrt{6.1} \approx 2.47 \]

Step 2: Calculate the angle θ

Find the angle θ using the arctangent function: \[ \theta = \arctan \left( \frac{y}{x} \right) \] Since the point (1.3, -2.1) is in the fourth quadrant, we need to adjust θ by adding 2π if θ is negative: \[ \theta = \arctan \left( \frac{-2.1}{1.3} \right) \approx -1.02 \text{ radians} \] Adjusting for the fourth quadrant: \[ \theta = 2\pi - 1.02 \approx 5.26 \text{ radians} \]

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane by specifying its distance from two perpendicular lines called axes. For any point, you need two values: the x-coordinate and the y-coordinate.

• The x-coordinate tells you how far the point is along the horizontal axis.
• The y-coordinate tells you how far the point is along the vertical axis.

For example, for the point (1.3, -2.1), 1.3 is the x-coordinate and -2.1 is the y-coordinate. This point is located 1.3 units to the right of the origin (0,0) and 2.1 units downwards. Understanding this is crucial before converting these coordinates into polar coordinates.

radius calculation

The radius in polar coordinates is the distance from the point to the origin. You can find this distance using the Pythagorean theorem.
The formula for the radius is:

• Square each coordinate value:

\( x^2 \) and \( y^2 \)

• Add the squares together: \( x^2 + y^2 \)
• Take the square root of the sum: \( \sqrt{x^2 + y^2} \)

\[ r = \sqrt{1.3^2 + (-2.1)^2} \approx 2.47 \]This calculation finds the length of the line directly connecting the origin to your point, which is the radius in polar coordinates.

angle determination

Determining the angle \(\theta\) involves finding the direction of the point relative to the positive x-axis. You calculate this using the arctangent function:
\( \theta = \arctan\left( \frac{y}{x} \right) \)
For our example point (1.3, -2.1), you find the angle as follows:

• Divide the y-coordinate by the x-coordinate:
\( \frac{-2.1}{1.3} \approx -1.615 \)

• Take the arctangent of this value:
\( \theta \approx -1.02 \text{ radians} \)

This represents the angle measured in radians from the positive x-axis to the line connecting the origin and the point.

quadrant adjustments

The angle calculation often requires adjustment depending on which quadrant the point is in. The Cartesian plane is divided into four quadrants:

• First Quadrant: Both x and y are positive.
• Second Quadrant: x is negative, y is positive.
• Third Quadrant: Both x and y are negative.
• Fourth Quadrant: x is positive, y is negative.

In our example, the point (1.3, -2.1) is in the fourth quadrant, where the angle from the positive x-axis is negative. To adjust the angle for correct polar coordinates:
\[ \theta = 2\pi - |\text{calculated angle}| \] \ Using the initial result: \[ \theta \approx 2\pi - 1.02 \approx 5.26 \text{ radians} \]
This adjustment ensures the angle represents the correct direction for a point in the fourth quadrant.