Problem 101 - Algebra and Trigonometry

Question

The rectangular coordinates of a point are given. Use a graphing utility in radian mode to find polar coordinates of each point to three decimal places. $$ (\sqrt{5}, 2) $$

Step-by-Step Solution

Answer

The polar coordinates of the point are (3, 0.588)

Step 1: Polar coordinates Conversion

To convert the rectangular coordinates to polar coordinates, two aspects are needed: the distance from the origin (r) and the angle (θ) formed by the point, the origin, and the positive x-axis. The formulas for their calculation are: \( r = \sqrt{x^2 + y^2} \) and \( θ = arctan(\frac{y}{x}) \) where x and y are the coordinates. In this case, \( x = \sqrt{5} \) and \( y = 2 \). Remember to use the graphing utility in radian mode for the calculation of θ.

Step 2: Calculate r

Substitute the values of x and y in the formula for r: \( r = \sqrt{(\sqrt{5})^2 + 2^2} \) which simplifies to: \( r = \sqrt{5 + 4} = \sqrt{9} = 3 \)

Step 3: Calculate θ

Substitute the values of x and y in the formula for θ: \( θ = arctan(\frac{2}{\sqrt{5}}) \). Using a graphing utility in radian mode, this calculation gives: \( θ = 0.588 \) radians to three decimal places.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to represent points in a two-dimensional plane using two numbers, usually denoted as \((x, y)\). These coordinates specify the horizontal distance \(x\) from the y-axis and the vertical distance \(y\) from the x-axis. This Cartesian system is the most common in mathematics because it aligns with our intuitive understanding of dimensions:

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

Here, the example given is the point \((\sqrt{5}, 2)\). The point is \(\sqrt{5}\) units right of the y-axis and 2 units up from the x-axis. To transfer this point into polar coordinates, we need to know it relative to an angle and distance from the origin, which requires understanding the next concepts.

Radian Mode

Understanding angles in radians is crucial when converting rectangular coordinates to polar coordinates. In many calculations, including those requiring trigonometric functions, using the radian mode is essential. Radians measure angles based on the radius of a circle:

• One radian is the angle formed when the arc length is equal to the circle's radius.
• It is a more natural measurement system in mathematics particularly for calculus and trigonometry.

In the context of our exercise, using a graphing utility in radian mode ensures that you calculate the angle \(\theta\) correctly when you apply the arctan function:\[ \theta = \arctan \left( \frac{y}{x} \right) \] For the point \((\sqrt{5}, 2)\), inverted calculation mistakes will be avoided by ensuring the correct mode (radian mode instead of degrees) is selected in the utility. This transforms a rectangle angle output into the polar form seamlessly.

Graphing Utility

A graphing utility can be a digital tool, like a calculator or software, that helps compute mathematical problems involving graphs and functions. These utilities streamline complex calculations and help visualize data and equations quickly:

• They come with built-in functions like trigonometric calculations, which are ideal for finding angles.
• By switching to radian mode, the utilities interpret angle-related outputs correctly, aligning them with many standard math problems.

In the case of converting rectangular coordinates to polar coordinates, a graphing utility helps establish the angle \(\theta\) necessary for the transformation. It's critical to ensure the utility is in radian mode; this ensures results like \(\theta = 0.588\) radians are quickly and accurately provided to three decimal places. Such precision is often crucial in mathematics and scientific calculations involving angles and distances, especially when situations demand accurate placement or movement such as in physics or engineering applications.