Problem 37 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (0,5) $$

Step-by-Step Solution

Answer

Polar coordinates: \((5, \frac{\pi}{2})\).

Step 1: Understand the Given Point

We are given a point in rectangular coordinates, represented as (0, 5). In rectangular coordinates, this means that the point is located at the origin moving 0 units along the x-axis and 5 units along the y-axis.

Step 2: Calculate the Radius (r)

To convert into polar coordinates, we need to find the distance from the origin to the point, represented by \( r \). The formula for calculating \( r \) is \( r = \sqrt{x^2 + y^2} \). Substitute \( x = 0 \) and \( y = 5 \) to get \( r = \sqrt{0^2 + 5^2} = \sqrt{25} = 5 \).

Step 3: Calculate the Angle (\( \theta \))

Next, we need to find the angle \( \theta \) formed with the positive x-axis. The formula for \( \theta \) when the point is in the first and second quadrants is \( \theta = \arctan(\frac{y}{x}) \). Since \( x = 0 \) and \( y = 5 \), the point is on the positive y-axis. Therefore, \( \theta \) is \( \frac{\pi}{2} \) because the entire angle to the positive y-axis is half of \( \pi \) radians.

Step 4: Construct the Polar Coordinates

Now that we have \( r = 5 \) and \( \theta = \frac{\pi}{2} \), we can write the polar coordinates as \((r, \theta)\), which yields \((5, \frac{\pi}{2})\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, provide a way to describe the location of a point in a plane using two values: an x-coordinate and a y-coordinate. These coordinates form a grid by overlaying horizontal and vertical number lines.
In rectangular coordinates:

• The x-coordinate indicates how far left or right the point is from the origin (0, 0).
• The y-coordinate indicates how far up or down the point is from the origin.

The point (0, 5) means the point is directly 5 units up along the y-axis, with no horizontal movement. Understanding this concept is crucial because it allows you to locate any point in space before converting it to another form, such as polar coordinates.
By viewing points using rectangular coordinates, it's easier to apply algebraic formulas to solve various mathematical problems involving geometry, physics, or engineering scenarios.

Radius Calculation

The radius in polar coordinates represents the distance from the origin (the center of the coordinate system) to a point. This is somewhat like finding the hypotenuse of a right triangle formed between the point, the origin, and the projection of the point onto the x-axis.
To calculate the radius, or the value of \( r \), you use the formula:\[ r = \sqrt{x^2 + y^2} \]This formula comes from the Pythagorean theorem.

• For our example, substitute \( x = 0 \) and \( y = 5 \).
• Calculate as follows: \( r = \sqrt{0^2 + 5^2} = \sqrt{25} = 5 \).

The result, \( r = 5 \), confirms our point is 5 units away from the origin, up along the y-axis. This value is essential when defining the position in polar coordinates.

Angle Calculation

Angle calculation in polar coordinates involves determining the direction of the radius line, starting from the positive x-axis. This angle is denoted by \( \theta \), measured in radians for mathematical precision.
For a point located at (0, 5), the point lies exactly on the positive y-axis. To find \( \theta \), consider the general angular formula involving arctan:\[ \theta = \arctan\left(\frac{y}{x}\right) \]However, if \( x = 0 \), this equation isn't necessary since standard trigonometric relationships tell us the angle.

• On the positive y-axis, the angle \( \theta \) is precisely \( \frac{\pi}{2} \), equivalent to 90 degrees.

This angle identity is fundamental as it represents the direction component of polar coordinates. Thus, a full polar conversion involves integrating the calculated radius and the angle together, ultimately crafting the polar coordinates as \((5, \frac{\pi}{2})\).