Problem 47 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Find polar coordinates of each point. Express \(\theta\) in radians. $$ (5,0) $$

Step-by-Step Solution

Answer

The polar coordinates of the point (5,0) are (5, 0)

Step 1: Calculate r

Calculate the radial coordinate using the formula r = sqrt(x² + y²), where x = 5 and y = 0. Substituting these values in gives \( r = sqrt(5² + 0²) = sqrt(25) = 5 \)

Step 2: Calculate θ

Calculate the polar angle using the formula θ = arctan(y/x), where x = 5 and y = 0. However, since y is zero, the arctan is not defined. In this case, we use a knowledge of the unit circle to determine that θ = 0 when x is positive and y is zero.

Step 3: Express in required form

The polar coordinates are thus (r, θ) = (5, 0), and since the problem requires θ to be in radians, there are no changes needed as 0 is the same in both degrees and radians.

Key Concepts

Rectangular Coordinates

In a two-dimensional plane, points are often represented using rectangular coordinates. These include two values: the

• horizontal position, denoted as \( x \)
• vertical position, denoted as \( y \)

Together, these values form an ordered pair, \((x, y)\). This method of position representation is also known as Cartesian coordinates, named after the mathematician René Descartes. For example, a point \((5, 0)\) indicates a position five units to the right of the vertical axis and exactly on the horizontal axis.
Rectangular coordinates are widely used because they align with the grid-like structure of paper or computer screens, making them intuitive for many users. However, they may become cumbersome when working with circular or angular figures, like circles and rotations.

Conversion to Polar Coordinates

Polar coordinates offer an alternative way to express the location of a point in a plane. They utilize two components:

• \( r \) for the radial distance from the origin
• \( \theta \) for the angle from the positive x-axis

In polar form, a point is represented as \((r, \theta)\).
To convert from rectangular coordinates \((x, y)\) to polar coordinates, we use the following steps:

• Calculate \( r \) using the formula \( r = \sqrt{x^2 + y^2} \). This computes the distance from the origin to the point.
• Find \( \theta \) with \( \theta = \arctan\left(\frac{y}{x}\right) \). This determines the angle that the line connecting the origin to the point makes with the positive x-axis.

When working with cases where \( y = 0 \), as in the point \((5, 0)\), agreement with known angle definitions is necessary. Here, \( \theta \) is 0 radians because the point lies on the positive x-axis and no vertical change exists.

Radians

Radians provide a way to measure angles based on the radius of a circle. An entire circle is considered to have \(2\pi\) radians, correlating directly with the circumference of the circle. By definition, one radian is the angle formed when the arc length equals the radius of the circle. This approach to angle measurement is often advantageous in mathematical calculations because it simplifies formulas in trigonometry and calculus.
In the context of our exercise, the angle \( \theta \) is the angle measurement used when expressing polar coordinates. When a point lies directly on the x-axis, as with point \((5, 0)\), the angle is 0 radians because it involves no rotation from the positive x-axis direction.
Recognizing angles in radians is important for mathematical conventions used in fields such as physics and engineering, where radian measure provides mathematical elegance and precision.