Problem 12 - Precalculus

Question

Convert each point to exact polar coordinates. Assume that \(0 \leq \theta<2 \pi.\) $$(3,-3)$$

Step-by-Step Solution

Answer

The polar coordinates are \((3\sqrt{2}, \frac{7\pi}{4})\).

Step 1: Understanding Polar Coordinates

Polar coordinates express a point in the plane using two values: 1. The radius ( ) - the distance from the origin to the point. 2. The angle ( heta) - the angle formed from the positive x-axis to the line segment connecting the origin to the point. We need to convert the Cartesian coordinates (3, -3) into this form.

Step 2: Calculate the Radius

The radius is calculated using the Pythagorean theorem. For a point (x, y), it's expressed as \[ r = \sqrt{x^2 + y^2} \]Substituting the given values:\[ r = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]Hence, the radius, \( r = 3\sqrt{2} \).

Step 3: Calculate the Angle

The tangent of the angle \( heta \) is given by \( an(\theta) = \frac{y}{x} \). Substitute the given (x, y):\( an(\theta) = \frac{-3}{3} = -1 \)The angle whose tangent is -1 is \( \frac{7\pi}{4} \) in the fourth quadrant (since x is positive and y is negative).

Step 4: Write the Polar Coordinates

Now we know both the radius \( r = 3\sqrt{2} \) and the angle \( \theta = \frac{7\pi}{4} \). Therefore, the polar coordinates for (3,-3) are:\( (3\sqrt{2}, \frac{7\pi}{4}) \)

Key Concepts

Cartesian Coordinates

Cartesian coordinates are widely used in geometry, and they represent points on the plane using pairs of numbers. Each point is defined by a pair (x, y), where:

• x is the horizontal distance from the origin (0,0), also known as the x-coordinate.
• y is the vertical distance from the origin, known as the y-coordinate.

For example, the point (3, -3) indicates that the location is 3 units to the right of the y-axis and 3 units downward. In a Cartesian coordinate system, this forms a simple and intuitive way to describe locations based on horizontal and vertical distances.
However, in some cases, polar coordinates may better describe a point due to their circular nature. Understanding Cartesian coordinates provides a foundation for converting these points into other forms, like polar coordinates.

Radius Calculation

The radius in polar coordinates measures the distance from the origin to the point on the plane. To convert from Cartesian to polar coordinates, we start by calculating this distance using the Pythagorean theorem.To find the radius, you use the formula:\[ r = \sqrt{x^2 + y^2} \]This formula comes from the idea of the hypotenuse in a right triangle formed by the point and the x and y axes.
For the point (3, -3), we calculate:

• First, square the x and y values: \( 3^2 = 9 \) and \((-3)^2 = 9 \).
• Add them together: \( 9 + 9 = 18 \).
• Take the square root to find the radius: \( r = \sqrt{18} = 3\sqrt{2} \).

This gives a concise view of how radius reflects the point's distance from the origin.

Angle Calculation

Calculating the angle is crucial in converting a Cartesian coordinate into a polar coordinate system. It involves determining the direction of the point relative to the positive x-axis. We use the arctangent function, which finds the angle whose tangent is a given number.The formula used is:\[ \tan(\theta) = \frac{y}{x} \]For the coordinates (3, -3):

• Divide y by x: \( \tan(\theta) = \frac{-3}{3} = -1 \).
• Find the arctangent of -1, which is \( \frac{7\pi}{4} \) radians in the fourth quadrant. This is because the x-component is positive, and the y-component is negative.

Hence, the angle \( \theta \) not only shows which quadrant the point rests in, but it also gives the rotational direction of the point from the positive x-axis. By understanding this angle, you complete the transformation to polar coordinates.