Problem 27 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ (6, \arctan (2)) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \(\left(\frac{6\sqrt{5}}{5}, \frac{12\sqrt{5}}{5}\right)\).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle (direction from the origin). Here, \(r = 6\) and \(\theta = \arctan(2)\).

Step 2: Use the Rectangular Coordinate Formulas

Rectangular coordinates \((x, y)\) can be found using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). We will apply these formulas to convert the given polar coordinates into rectangular coordinates.

Step 3: Calculate the \(x\)-coordinate

To find \(x\), use \(x = 6 \cos(\arctan(2))\). Since \(\arctan(2)\) means that in a right triangle, the opposite side to the angle is 2 and the adjacent side is 1, the hypotenuse will be \(\sqrt{2^2 + 1^2} = \sqrt{5}\). Thus, \(\cos(\theta) = \frac{1}{\sqrt{5}}\). Substitute to get \(x = 6 \times \frac{1}{\sqrt{5}} = \frac{6}{\sqrt{5}}\).

Step 4: Calculate the \(y\)-coordinate

To find \(y\), use \(y = 6 \sin(\arctan(2))\). With \(\arctan(2)\), where opposite = 2 and adjacent = 1, we have \(\sin(\theta) = \frac{2}{\sqrt{5}}\). Thus, \(y = 6 \times \frac{2}{\sqrt{5}} = \frac{12}{\sqrt{5}}\).

Step 5: Simplify the Rectangular Coordinates

Now we have \(x = \frac{6}{\sqrt{5}}\) and \(y = \frac{12}{\sqrt{5}}\). For simplicity in calculations or presentation, these can be multiplied by \(\sqrt{5}/\sqrt{5}\) to rationalize the denominator if preferred. So, \(x = \frac{6\sqrt{5}}{5}\) and \(y = \frac{12\sqrt{5}}{5}\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two components: the radial distance from the origin and the angular direction from the positive x-axis. These two components are typically denoted as \((r, \theta)\), where \(r\) is the distance from the origin to the point, often called the radius, and \(\theta\) is the angle measured in radians or degrees between the positive x-axis and the line segment connecting the origin to the point.

Unlike rectangular (or Cartesian) coordinates, polar coordinates offer an efficient way to describe locations that are naturally circular or rotational in nature. This makes them particularly useful in fields like engineering and physics, where rotational symmetry occurs regularly. For our problem, the point is represented as \((6, \arctan(2))\), indicating a radius of 6 and an angle whose tangent is 2.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, express a point by its distances from two perpendicular axes intersecting at a point called the origin. These coordinates are given in the form \((x, y)\), where \(x\) and \(y\) represent the horizontal and vertical distances from the origin, respectively.

This coordinate system is extremely intuitive for most everyday applications, as it mirrors familiar dimensions of everyday space. Rectangular coordinates make graphing and calculations straightforward and are the basis for many tools and technologies in both mathematics and physical sciences. In our exercise, the conversion from polar coordinates using methods like trigonometry allows us to translate the point into this familiar \((x, y)\) form.

Coordinate Conversion

To convert from polar to rectangular coordinates, we utilize the relationship between the radius \(r\), the angle \(\theta\), and the coordinate points \((x, y)\). This is done through the formulas:

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

These formulas derive from trigonometric definitions and rest on the geometric fact that any point described in polar coordinates can be projected onto the x and y axes.

In the given exercise, with \(r = 6\) and \(\theta = \arctan(2)\), we determined that \(\cos(\theta)\) corresponds to \(\frac{1}{\sqrt{5}}\) and \(\sin(\theta)\) to \(\frac{2}{\sqrt{5}}\) based on the right triangle known from the tangent function.

Trigonometric Functions

Trigonometric functions are essential in converting between polar and rectangular coordinates because they define the relationships between the angles and sides of right triangles. Some of the primary trigonometric functions include sine, cosine, and tangent.

For the angle \(\theta = \arctan(2)\), the tangent of \(\theta\) is the ratio of the opposite side to the adjacent side in a right triangle, which equals 2. Knowing this, we set up a right triangle where the opposite side is 2 and the adjacent side is 1, leading us to calculate the hypotenuse as \(\sqrt{5}\). This triangle configuration allows us to find:

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

These ratios then allow us to use the sine and cosine values to find precise rectangular coordinates, showing how trigonometry aids in coordinate conversion.