Problem 31 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (5,5 \pi) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-5, 0)\).

Step 1: Polar to Rectangular Formula Reminder

The polar coordinates \((r, \theta)\) can be converted to rectangular coordinates \((x, y)\) using the formulas: \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]Here, \(r = 5\) and \(\theta = 5\pi\).

Step 2: Calculate the x-coordinate

Plug the values into the formula for the x-coordinate:\[ x = 5 \cos(5\pi) \]We know that \(\cos(5\pi) = -1\) because the cosine of \(\pi\), \(3\pi\), etc., alternates between -1. Therefore, \[ x = 5 \times (-1) = -5 \]

Step 3: Calculate the y-coordinate

Now, we'll calculate the y-coordinate using the given polar values:\[ y = 5 \sin(5\pi) \]We know that \(\sin(5\pi) = 0\) because the sine of any multiple of \(\pi\) is zero. Therefore, \[ y = 5 \times 0 = 0 \]

Step 4: Finalize the Rectangular Coordinates

Combine the calculated \(x\) and \(y\) coordinates to express the solution in the rectangular form:The rectangular coordinates are \((-5, 0)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools when dealing with conversions between polar and rectangular coordinates. They help in relating angles to the circle's radius. The most used trigonometric functions in this context are sine (\(\sin\)) and cosine (\(\cos\)). These functions are crucial in determining the horizontal and vertical components of a point when moved from the polar coordinate system to the rectangular system.

Here's how these functions work in conversions:

• Cosine (\(\cos\)): Used to determine the x-coordinate. It measures the horizontal distance from the origin to the point as you project the radius onto the x-axis.
• Sine (\(\sin\)): Used to determine the y-coordinate. It measures the vertical distance from the origin to the point as you project the radius onto the y-axis.

By thinking of the polar coordinates as points on a circle centered at the origin, trigonometric functions allow the conversion between these systems by leveraging the circle's natural properties.

Coordinate Conversion

Coordinate conversion is the process of changing a point's description from a polar system to a rectangular one, or vice versa. This is especially useful in fields like engineering and physics, where different problems are more efficiently solved in one coordinate system over the other.

The formula for conversion from polar \((r, \theta)\) to rectangular \((x, y)\) coordinates uses:

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

By applying these formulas, the distance and angle from the polar coordinates are used to calculate the horizontal \(x\) and vertical \(y\) positions in a rectangular setting.

In our example, given \((r, \theta) = (5, 5\pi)\), the x and y coordinates are calculated by evaluating the values of \(\cos(5\pi)\) and \(\sin(5\pi)\), resulting in \((-5, 0)\). This transformation helps visualize points in the standard Cartesian plane.

Angle Measurement

Angle measurement is a fundamental concept when working with polar coordinates, as it determines the direction of the point from the origin. In polar coordinates, the angle \(\theta\) is measured in radians, a unit favored in trigonometry for its natural relation to circles.

To understand this better, consider the full circle:

• A circle has 360 degrees, which equates to \(2\pi\) radians.
• An angle's positioning, like \(5\pi\), implies rotation about the circle. Here, \(5\pi\) means the angle has gone around the circle more than twice, landing effectively back on the negative x-axis.

This understanding of angles is pivotal for conversions because knowing where \(\theta\) lands on the circle determines the sine and cosine values used in transforming polar coordinates to rectangular ones. For our exercise, the correct application of these principles leads to precisely placing the resultant rectangular coordinates at \((-5, 0)\).