Problem 36 - College Trigonometry

Question

Convert the point from polar coordinates into rectangular coordinates. $$ \left(13, \arctan \left(\frac{12}{5}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (5, 12).

Step 1: Identify Polar Coordinates

The given polar coordinates are \((r, \theta) = (13, \arctan(\frac{12}{5}))\) where the radial coordinate \(r = 13\) and the angular coordinate \(\theta = \arctan(\frac{12}{5})\).

Step 2: Determine \(\theta\) in Radians

The angle \(\theta\) is given as \(\arctan(\frac{12}{5})\). This means \(\tan(\theta) = \frac{12}{5}\). The value of \(\theta\) can be interpreted as the angle whose tangent is \(\frac{12}{5}\).

Step 3: Calculate Rectangular Coordinates

To convert from polar to rectangular coordinates, use the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\). First, calculate \(\cos(\theta)\) and \(\sin(\theta)\) using the known tangent: \(\tan(\theta) = \frac{12}{5}\). By the Pythagorean Identity, you can construct a right triangle with opposite side 12, adjacent side 5, and hypotenuse 13. Thus, \(\cos(\theta) = \frac{5}{13}\) and \(\sin(\theta) = \frac{12}{13}\).

Step 4: Substitute and Compute \(x\) and \(y\)

Now substitute into the formulas: \(x = 13 \cos(\theta) = 13 \times \frac{5}{13} = 5\), and \(y = 13 \sin(\theta) = 13 \times \frac{12}{13} = 12\). This gives the rectangular coordinates \((x, y) = (5, 12)\).

Key Concepts

Coordinate Conversion

Converting between polar and rectangular coordinates is an essential skill in mathematics, specifically in trigonometry and calculus. The conversion involves transforming the representation of a point on a plane.

In polar coordinates, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. To convert a point from polar to rectangular coordinates, you use the formulas:

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

In the exercise, the given polar coordinates are \((13, \arctan(\frac{12}{5}))\). The goal was to convert this to rectangular coordinates \((x, y)\).

When performing such conversions, accuracy in computing trigonometric values is vital to correctly find the x and y coordinates. This fundamental conversion underlies many spatial calculations and can be verified using a coordinate graph.

Right Triangle Trigonometry

Understanding right triangle trigonometry is crucial when dealing with trigonometric functions. A right triangle consists of one 90-degree angle, and the other two angles are complementary. These angles determine the ratios of the sides of the triangle.

For the exercises we're considering, the key trigonometric functions are sine, cosine, and tangent:

• Sine (\(\sin\)): Opposite/Hypotenuse
• Cosine (\(\cos\)): Adjacent/Hypotenuse
• Tangent (\(\tan\)): Opposite/Adjacent

In this exercise, the tangent of angle \(\theta\) is known as \(\frac{12}{5}\). This means that in a right triangle that represents our problem, the opposite side is 12 units, and the adjacent side is 5 units.

To find the sine and cosine values for \(\theta\), which are used in coordinate conversion, we can relate these side lengths back to the triangle's hypotenuse. This is where our final concept comes into play.

Pythagorean Identity

The Pythagorean Identity stems from the Pythagorean Theorem, a fundamental principle in geometry relating the sides of a right triangle:
\[ a^2 + b^2 = c^2 \]
where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.

In trigonometry, this identity translates into the equation:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This identity is applied here by first finding the hypotenuse using the ratios from the triangle's sides. If the opposite and adjacent sides of the triangle are 12 and 5 respectively, the hypotenuse is calculated as 13 using \(12^2 + 5^2 = 13^2\).

From this complete right triangle, \(\cos \theta\) can be found as \(\frac{5}{13}\), and \(\sin \theta\) as \(\frac{12}{13}\). These sin and cos values contribute to converting polar coordinates to rectangular, shown in the solution by transforming \(x\) and \(y\) coordinates accurately.