Problem 69 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (8.3,4.2) $$

Step-by-Step Solution

Answer

(9.3, 0.47)

Step 1 - Understand the Relationship

The polar coordinates \((r, \theta)\) of a point are related to its rectangular coordinates \((x, y)\) by the formulas:\[ r = \sqrt{x^2 + y^2} \]\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Step 2 - Calculate the Radius

Substitute \(x = 8.3\) and \(y = 4.2\) into the formula for \(r\):\[ r = \sqrt{8.3^2 + 4.2^2} = \sqrt{68.89 + 17.64} = \sqrt{86.53} \approx 9.3 \]

Step 3 - Determine the Angle

Use the formula for \(\theta\) by substituting \(x = 8.3\) and \(y = 4.2\):\[ \theta = \tan^{-1}\left(\frac{4.2}{8.3}\right) \approx \tan^{-1}(0.506) \]Using a calculator, this gives \(\theta \approx 0.47 \, radians \).

Step 4 - Combine Results

The polar coordinates \((r, \theta)\) are \((9.3, 0.47)\).

Key Concepts

rectangular coordinates

To start, let’s recall what rectangular coordinates (or Cartesian coordinates) are. They define the position of a point in a 2D plane using two values: an x-coordinate (horizontal axis) and a y-coordinate (vertical axis). In this particular problem, our rectangular coordinates are given as (8.3, 4.2).
These values indicate that the point is 8.3 units to the right of the origin along the x-axis and 4.2 units up along the y-axis.
Knowing this helps us understand where the point lies on a standard graph.

radius calculation

The radius in polar coordinates is the distance from the origin to the point. Think of it as the hypotenuse of a right-angled triangle where the point and the origin form the vertices, and the sides of the triangle are aligned with the x-axis and y-axis.
To find the radius, we use the Pythagorean theorem. We need to find the square root of the sum of the squares of the x-coordinate and the y-coordinate:
\[\begin{equation} r = \sqrt{x^2 + y^2} \. \end{equation}\] Inserting our values, we get:
\[ r = \sqrt{8.3^2 + 4.2^2} = \sqrt{68.89 + 17.64} = \sqrt{86.53} \approx 9.3 \]
So, the radius is approximately 9.3 units.

angle determination

The next step is to calculate the angle, \( \theta \), which tells us the direction from the origin to the point. We can find this angle using the formula:
\[ \theta = \tan^{-1}\left( \frac{y}{x} \right)\]
For this problem, we substitute our x and y values:
\[ \theta = \tan^{-1}\left( \frac{4.2}{8.3} \right) \approx \tan^{-1}(0.506)\]
Using a calculator, this gives an angle of approximately 0.47 radians. In real-world scenarios, angles might be converted to degrees, but in this exercise, we will keep it in radians.

coordinate transformation

Finally, let's put it all together. After calculating the radius and angle, we can now express the original rectangular coordinates (8.3, 4.2) in polar coordinates.
The polar coordinates combine the radius \(r\) with the angle \( \theta \) to describe the position of the point in the plane.
For our example, the polar coordinates are \( (9.3, 0.47) \).
Remember:

• Radius \( r \) is the distance from the origin.
• Angle \( \theta \) is measured in radians (or degrees) from the positive x-axis.

Converting between these coordinate systems is a powerful tool for solving many geometry and physics problems.
It allows us to transform complex problems into simpler ones by choosing the most suitable coordinate system.