Problem 54 - Precalculus with Limits

Question

In Exercises 37-54, a point in rectangular coordinates is given. Convert the point to polar coordinates. \(\left(7, 15\right)\)

Step-by-Step Solution

Answer

The polar coordinates for the point (7,15) in rectangular coordinates are approximately \((\sqrt{274} , θ)\) and θ is gotten from step 2 with actual measure depending on the relevant quadrant.

Step 1: Compute 'r'

Use the Cartesian to Polar Conversion formula: \(r=\sqrt{x^{2}+y^{2}}\). By substituting the rectangular system values (7, 15), we get: \(r=\sqrt{7^{2}+15^{2}} = \sqrt{49+225} = \sqrt{274}\)

Step 2: Calculate 'θ'

Compute θ by the formula: \(θ =tan^{-1}(y/x)\). By substituting the rectangular system values (7, 15), we get: \(θ =tan^{-1}(15/7)\)

Step 3: Convert 'θ' to degrees

To convert θ to degrees θdeg, multiply the result gotten from step 2 with \(180/π\). Hence, \(θ_{deg} = θ * (180/π)\)

Step 4: Choose the coordinate

The polar coordinate convention (r, θ) includes the counterclockwise direction as positive for the angle measure. So if the angle is in quadrant 1 or 4, it is considered positive. If it lies in quadrant 2 or 3, it is added with 180 degree.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way to specify the position of a point in a two-dimensional plane. The coordinates are denoted as \(x, y\) and are based on the Cartesian coordinate system. In this system:

• The x-coordinate specifies the horizontal distance from the origin, which is the intersection of the x-axis and y-axis.
• The y-coordinate represents the vertical distance from the origin.

The origin itself is denoted as \(0, 0\). The x-axis runs horizontally, and the y-axis runs vertically, dividing the plane into four quadrants. Understanding how these coordinates map out in the plane is essential for converting to different coordinate systems, such as polar coordinates.

Conversion Formulas

Conversion formulas are the mathematical expressions used to switch between different types of coordinate systems. When converting from rectangular coordinates (x, y) to polar coordinates \(r, \theta\), we use the formulas:

• For finding the radius \(r\): \[r=\sqrt{x^{2}+y^{2}}\].
• To calculate the angle \(\theta\): \[\theta = \tan^{-1}\left(\frac{y}{x}\right)\].

These formulas are derived from the basic principles of right-angle triangle trigonometry. Here, \(r\) represents the distance from the origin to the point, which can be viewed as the hypotenuse of a right triangle formed with x and y as the other two sides. The angle \(\theta\) indicates the direction of the point, measured in degrees or radians from the positive x-axis.

Quadrants

Quadrants are the four sections created by the intersection of the x-axis and y-axis on the Cartesian plane. They are important in determining the sign and angle for points when converting to polar coordinates. The quadrants are numbered counterclockwise:

• Quadrant I: Both x and y are positive (top-right).
• Quadrant II: x is negative, y is positive (top-left).
• Quadrant III: Both x and y are negative (bottom-left).
• Quadrant IV: x is positive, y is negative (bottom-right).

When converting rectangular coordinates to polar coordinates, acknowledging which quadrant the point is in helps properly assign the angle \(\theta\). If the calculated angle does not align with the quadrant of the original point, adjustments such as adding 180° might be necessary to ensure the angle is correct.

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles in triangles. In the context of converting rectangular coordinates to polar coordinates, trigonometry helps us find the angle \(\theta\) and distance \(r\).

• The tangent function, represented as \(\tan^{-1\left(\frac{y}{x}\right)}\), is primarily used to find the angle based on the ratio of the y-coordinate to the x-coordinate.
• The Pythagorean theorem, with the formula \[r=\sqrt{x^{2}+y^{2}}\], allows us to calculate the hypotenuse or the distance of the point from the origin using x and y values.

These trigonometric principles simplify the transition between coordinate systems by providing tools to interpret points relative to the origin and determine their position in a coherent way.