Problem 48 - College Trigonometry

Question

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

Step-by-Step Solution

Answer

The polar coordinates are approximately \((5, 1.107)\).

Step 1: Understand the Problem

We are given the rectangular coordinates \((x, y) = (\sqrt{5}, 2\sqrt{5})\) and need to convert them to polar coordinates \((r, \theta)\). The polar coordinates are defined by \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\).

Step 2: Calculate the Radius \(r\)

Using the formula \(r = \sqrt{x^2 + y^2}\), substitute the given values. \[r = \sqrt{(\sqrt{5})^2 + (2\sqrt{5})^2} = \sqrt{5 + 4 \times 5} = \sqrt{5 + 20} = \sqrt{25} = 5\]

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

Using the formula \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\), substitute the given values:\[\theta = \tan^{-1}\left(\frac{2\sqrt{5}}{\sqrt{5}}\right) = \tan^{-1}(2)\] Since \(\tan^{-1}(2)\) doesn't correspond to a special angle, use a calculator to find the angle in radians.

Step 4: Verify the Quadrant

Ensure the angle \(\theta\) is in the correct quadrant. The original point has positive \(x\) and \(y\) values, which places it in the first quadrant where angles range from \(0\) to \frac{\pi}{2}. This confirms the calculated \(\theta\) is correct as \tan^{-1}(2)\ is about 1.107 radians, which lies in the first quadrant.

Step 5: State the Polar Coordinates

The polar coordinates \((r, \theta)\) for the given rectangular coordinates are \((5, \tan^{-1}(2))\). In radians, this is approximately \((5, 1.107)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are an essential concept in the world of mathematics, especially in geometry and physics. This coordinate system, also known as Cartesian coordinates, represents points in a plane using two values: the x-coordinate and the y-coordinate. These values describe how far along the horizontal axis (x) and vertical axis (y) a point is positioned from the origin \((0,0)\).
For example, if you have the point \((\sqrt{5}, 2\sqrt{5})\), it translates to going \(\sqrt{5}\) units right from the origin and \(2\sqrt{5}\) units up. Rectangular coordinates make plotting straightforward and are incredibly useful in graphing and analysis.
Understanding rectangular coordinates is the first step in transforming them into other representations, such as polar coordinates. This conversion is particularly valuable when dealing with problems involving rotational symmetry or angles.

Radius Calculation

The first component of polar coordinates is the radius, denoted as \(r\). The radius measures the distance of a point from the origin in a polar coordinate system, which is distinct from the straight paths defined in rectangular coordinates.
To calculate \(r\), you can use the formula:

• \(r = \sqrt{x^2 + y^2}\)

This equation comes from the Pythagorean Theorem, as it relates the sides of a right triangle to its hypotenuse, which in this context is the radius. For instance, applying the coordinates \((\sqrt{5}, 2\sqrt{5})\), you substitute and compute:

• \(r = \sqrt{(\sqrt{5})^2 + (2\sqrt{5})^2} = 5\)

This indicates that the point is 5 units away from the origin in the polar system, providing a straightforward measure of its location relative to the center of the circle.

Angle Calculation

The second crucial element in polar coordinates is the angle, represented as \(\theta\). This angle signifies the direction from the origin to the point, measured counterclockwise from the positive x-axis.
You calculate \(\theta\) using the tangent inverse function, which involves the coordinates:

• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

Inserting our point \((\sqrt{5}, 2\sqrt{5})\) into the formula gives us:

• \(\theta = \tan^{-1}\left(\frac{2\sqrt{5}}{\sqrt{5}}\right) = \tan^{-1}(2)\)

Since \(\tan^{-1}(2)\) produces an angle not linked to a standard trigonometric angle, you would typically use a calculator to find \(\theta\). Here, it evaluates to about 1.107 radians. This calculation helps locate the point's direction precisely and is critical in situating the point correctly on the polar graph.

Quadrant Verification

Verifying the quadrant is essential to ensure the point's angle \(\theta\) is computed correctly and consistently with its position in rectangular coordinates. Each quadrant in the Cartesian plane has specific characteristics based on the sign of the x and y coordinates.
For the point \((\sqrt{5}, 2\sqrt{5})\):

• The x-coordinate \(\sqrt{5}\) is positive.
• The y-coordinate \(2\sqrt{5}\) is also positive.

These positive values indicate that the point is located in the First Quadrant, where both coordinates are positive and angles range from 0 to \(\frac{\pi}{2}\) radians. The calculated angle of approximately 1.107 radians matches this quadrant perfectly.
Performing a quadrant verification confirms that each calculation of \(\theta\) aligns with the expected position on the coordinate graph. This crucial step reduces errors and provides confidence in the conversion process.