Problem 47 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ (6,8) $$

Step-by-Step Solution

Answer

The polar coordinates are \((10, 0.93)\).

Step 1: Identify the Coordinates

The point given in rectangular coordinates is (6, 8). This means the x-coordinate is 6 and the y-coordinate is 8.

Step 2: Calculate the Radius

The radius \( r \) in polar coordinates can be found using the formula \( r = \sqrt{x^2 + y^2} \). Substitute the given values: \( r = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \).

Step 3: Calculate the Angle

The angle \( \theta \) in polar coordinates can be found using the formula \( \theta = \tan^{-1}\left( \frac{y}{x} \right) \). Substitute the given values: \( \theta = \tan^{-1}\left( \frac{8}{6} \right) = \tan^{-1}\left( \frac{4}{3} \right) \).

Step 4: Adjust the Angle for Range

Since the point (6, 8) lies in the first quadrant where both x and y are positive, the angle \( \theta \) is already in the correct range [0, 2\pi). Using a calculator, \( \theta \approx 0.93 \) in radians.

Step 5: Write the Polar Coordinates

The polar coordinates for the point (6, 8) are \((10, 0.93)\).

Key Concepts

Rectangular Coordinates

In algebra and calculus, understanding coordinate systems is crucial, particularly when dealing with points in a plane. Rectangular coordinates, also known as Cartesian coordinates, are represented by the pair \(x, y\). Here, \(x\) is the horizontal position, and \(y\) is the vertical position of a point on a plane. The rectangular coordinate system divides a plane into four quadrants, determined by the signs of \(x\) and \(y\):

• First Quadrant: both \(x\) and \(y\) are positive.
• Second Quadrant: \(x\) is negative, \(y\) is positive.
• Third Quadrant: both \(x\) and \(y\) are negative.
• Fourth Quadrant: \(x\) is positive, \(y\) is negative.

This system allows for precise plotting of points, which can then be transformed into other coordinate systems, such as polar coordinates, for various applications in math and science.

Conversion Process

Converting from rectangular to polar coordinates involves two main steps: calculating the radius and the angle.

• **Radius Calculation**: The polar radius \(r\) is found using the Pythagorean theorem. It is the distance from the point to the origin \( (0, 0) \) and calculated as \(r = \sqrt{x^2 + y^2}\).
• **Angle Calculation**: The angle \(\theta\) represents the direction of the point from the origin, calculated as \(\theta = \tan^{-1}\left( \frac{y}{x} \right)\). **Important note:** The angle needs to be verified or adjusted depending on the quadrant in which the point lies to ensure \(0 \leq \theta < 2\pi\).

Both of these components together form the new polar coordinate \( (r, \theta) \). This conversion is invaluable because polar coordinates offer a unique perspective in dynamics, electromagnetics, and many fields where circular motion is present.

Trigonometric Functions

The conversion between coordinate systems often involves trigonometric functions, critical in examining circular and periodic phenomena. These functions include sine, cosine, and tangent.

• **Sine and Cosine**: Directly relate angles to ratios of sides of right triangles, useful for converting between different representations in coordinate systems.
• **Tangent**: Especially useful when calculating angles, as in these conversions. Its inverse, the arctan function \(\tan^{-1}\), helps find angles when given side lengths, specifically used to compute \(\theta\) in this type of conversion.

When converting to polar coordinates, trigonometric functions allow us to manipulate and understand the geometry of the point relative to the origin efficiently. By thoroughly understanding these functions, one gains the tools necessary to transition fluidly between rectangular and polar systems, a foundational skill in advanced mathematics.