Problem 31 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(2, \pi-\arctan \left(\frac{1}{2}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-\frac{4 \sqrt{5}}{5}, \frac{2 \sqrt{5}}{5})\).

Step 1: Understanding Polar Coordinates

The given polar coordinates are \( (r, \theta) = (2, \pi - \arctan(\frac{1}{2})) \). In polar coordinates, \( r \) is the radius (distance from the origin) and \( \theta \) is the angle from the positive x-axis.

Step 2: Formulating Conversion Equations

To convert polar coordinates to rectangular coordinates, use the formulas:\[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) \]

Step 3: Simplifying the Angle

First, find \( \theta = \pi - \arctan(\frac{1}{2}) \). Recognize that \( \arctan(\frac{1}{2}) \) gives an angle whose tangent is \( \frac{1}{2} \), and \( \pi - \arctan(\frac{1}{2}) \) places this angle in the second quadrant.

Step 4: Calculating \( \cos(\theta) \) and \( \sin(\theta) \)

Use trigonometric identities for \( \theta = \pi - \phi \):\[ \cos(\pi - \phi) = -\cos(\phi) \quad \text{and} \quad \sin(\pi - \phi) = \sin(\phi) \]where \( \phi = \arctan(\frac{1}{2}) \). Since \( \tan(\phi) = \frac{1}{2} \), we can think of \( \phi \) as being in a right triangle with opposite side 1 and adjacent side 2.

Step 5: Using the Right Triangle

From the right triangle, use the Pythagorean theorem to find the hypotenuse: \[ ext{Hypotenuse} = \sqrt{1^2 + 2^2} = \sqrt{5} \].Thus, \( \cos(\phi) = \frac{2}{\sqrt{5}} \) and \( \sin(\phi) = \frac{1}{\sqrt{5}} \).

Step 6: Applying Trigonometric Values

Now, apply the angle identities:\[ \cos(\theta) = -\frac{2}{\sqrt{5}}, \quad \sin(\theta) = \frac{1}{\sqrt{5}} \].

Step 7: Finding Rectangular Coordinates

Substitute \( r = 2 \), \( \cos(\theta) = -\frac{2}{\sqrt{5}} \), and \( \sin(\theta) = \frac{1}{\sqrt{5}} \) into the conversion equations:\[ x = 2 \times \left(-\frac{2}{\sqrt{5}}\right) = -\frac{4}{\sqrt{5}}, \quad y = 2 \times \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}} \].

Step 8: Rationalizing the Denominators

Finally, multiply the numerator and denominator by \( \sqrt{5} \) to rationalize the denominators:\[ x = -\frac{4 \sqrt{5}}{5}, \quad y = \frac{2 \sqrt{5}}{5} \].

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical expressions that allow us to relate the angles and sides of triangles. They are crucial in many areas of mathematics, including geometry and calculus. In this problem, certain identities help convert polar coordinates to rectangular coordinates.

Key identities include those connecting sine and cosine functions. Specifically, when dealing with angles in the unit circle:

• For an angle \(\theta\), \(\cos(\theta)\) and \(\sin(\theta)\) describe cos and sin values at that angle respectively.
• Another useful identity used in this problem is for angles in the second quadrant, \(\cos (\pi - \phi) = -\cos(\phi)\) and \(\sin(\pi - \phi) = \sin(\phi)\).

By applying these identities, we find values for cos and sin that reflect their signs appropriately according to the quadrant on the unit circle.

Right Triangle

The concept of the right triangle is fundamental in trigonometry, primarily through the Pythagorean theorem. Right triangles have one 90-degree angle, and two legs perpendicular to each other. When solving for the conversion of polar coordinates, creating a right triangle helps in determining trigonometric values.

In our problem, the angle \(\phi\) is determined by \(\tan^{-1}(\frac{1}{2})\), which can be visualized as the angle in a right triangle where:

• The opposite side is 1.
• The adjacent side is 2.
• The hypotenuse calculated through the theorem \(\sqrt{1^2 + 2^2} = \sqrt{5}\).

Knowing the lengths of these sides allows us to determine:

• \(\cos(\phi) = \frac{2}{\sqrt{5}}\)
• \(\sin(\phi) = \frac{1}{\sqrt{5}}\)

Polar Coordinates

Polar coordinates provide a different way of locating a point in a plane, compared to the usual Cartesian coordinate system. Instead of using two perpendicular axes (x and y) for distance measurements, polar coordinates rely on a distance from the origin and an angular direction from a chosen reference line—typically the positive x-axis.

In the given problem, the initial point is presented as \( (r, \theta) = (2, \pi - \arctan(\frac{1}{2})) \). Here:

• \(r = 2\) represents the radial distance from the origin.
• \(\theta = \pi - \arctan(\frac{1}{2})\) represents the angle.

Polar coordinates offer a potent means of converting information from a rotational perspective to the linear perspectives offered by rectangular coordinates.

Rectangular Coordinates

Unlike polar coordinates, rectangular coordinates (also known as Cartesian coordinates) use two perpendicular lines to define the position of a point in a plane. These are often known as the x-axis and y-axis.

To convert from polar to rectangular coordinates, we use specific formulas:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

In solving the problem, substituting the known trigonometric identities and values:

• \(x = 2 \times \left(-\frac{2}{\sqrt{5}}\right) = -\frac{4}{\sqrt{5}}\) and converting it to \(-\frac{4 \sqrt{5}}{5}\)
• \(y = 2 \times \frac{1}{\sqrt{5}} = \frac{2}{\sqrt{5}}\) and converting it to \(\frac{2 \sqrt{5}}{5}\)

This transformation method exemplifies how polar coordinates are effectively converted into the more linear, Cartesian format.