Problem 59 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (3,0) $$

Step-by-Step Solution

Answer

(3, 0)

Step 1 - Understand the relationship

Understand that rectangular coordinates \(x, y\) can be converted to polar coordinates using the formulas \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan\left(\frac{y}{x}\right)\)

Step 2 - Calculate \(r\)

Given the point \(3,0\), calculate \(r\) using the formula: \[r = \sqrt{3^2 + 0^2} = \sqrt{9} = 3 \]

Step 3 - Calculate \(\theta\)

Determine \(\theta\) using the formula: \[\theta = \arctan\left(\frac{0}{3}\right) = \arctan(0) = 0 \]

Step 4 - Write the polar coordinates

Combine the results from Steps 2 and 3 to express the polar coordinates. Thus, the polar coordinates of the point \(3,0\) are \( (3, 0) \)

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use the \(x, y\) system to locate a point on a plane.
Picture a graph with two intersecting lines: the horizontal line is the x-axis, and the vertical line is the y-axis. The point where they intersect is called the origin \(0,0\).
To find where a point like \(3,0\) is located, you would move 3 units along the x-axis and 0 units along the y-axis.
In rectangular coordinates, the first number in the pair is the x-coordinate (horizontal position), and the second number is the y-coordinate (vertical position).

• x-axis: Horizontal line
• y-axis: Vertical line
• Origin: \(0,0\)

Understanding rectangular coordinates is essential because it forms the basis for converting to polar coordinates.

Conversion Formulas

To switch from rectangular coordinates (\(x, y\)) to polar coordinates (\(r, \theta\)), you need to know two important formulas.
The first formula helps you find the radial distance \(r\):
\[r = \sqrt{x^2 + y^2}\]
This formula uses the Pythagorean theorem to calculate the distance from the origin to the point.
The second formula helps you find the angle \(\theta\):
\[\theta = \arctan\left(\frac{y}{x}\right)\]
This formula finds the angle the point makes with the positive x-axis.

• For \(r\): Use the Pythagorean theorem
• For \(\theta\): Use the arctangent function

These two formulas are crucial steps to convert any set of rectangular coordinates into polar coordinates.

Trigonometry

Trigonometry helps us understand the relationships between the sides and angles of triangles.
In the conversion process, you'll often encounter trigonometric functions such as sine, cosine, and tangent.
For instance:

• Sine (sin): Ratio of the opposite side to the hypotenuse
• Cosine (cos): Ratio of the adjacent side to the hypotenuse
• Tangent (tan): Ratio of the opposite side to the adjacent side

In the formula \(\theta = \arctan\left(\frac{y}{x}\right)\), tangent is used to find the angle because it compares the vertical movement (y) to the horizontal movement (x).
When converting coordinates, these trigonometric principles allow you to translate a point's position into an angle.
Understanding these relationships helps to visualize and calculate the polar coordinates more easily.