Problem 19 - Contemporary Precalculus

Question

Convert the polar coordinates to rectangular coordinates. $$(-4,-\pi / 7)$$

Step-by-Step Solution

Answer

Question: Convert the polar coordinates \((-4, -\pi / 7)\) to rectangular coordinates. Answer: Approximately \((3.51, 1.90)\).

Step 1: Identify polar coordinates

The given polar coordinates are \(r = -4\) and \(\theta = -\pi / 7\).

Step 2: Convert r and θ to x

Use the polar-to-rectangular conversion formula for x: $$x = r \cos{\theta}$$ Plug in the given values, \(r = -4\) and \(\theta = -\pi / 7\): $$x = (-4) \cos{(-\pi / 7)}$$ Compute the value of x: $$x \approx 3.51$$

Step 3: Convert r and θ to y

Use the polar-to-rectangular conversion formula for y: $$y = r \sin{\theta}$$ Plug in the given values, \(r = -4\) and \(\theta = -\pi / 7\): $$y = (-4) \sin{(-\pi / 7)}$$ Compute the value of y: $$y \approx 1.90$$

Step 4: Write the rectangular coordinates

With the calculated x and y values, the rectangular coordinates are: $$(x, y) \approx (3.51, 1.90)$$ So the polar coordinates \((-4, -\pi / 7)\) are approximately equal to the rectangular coordinates \((3.51, 1.90)\).

Key Concepts

Polar Coordinates

Polar coordinates represent points in a two-dimensional space using a different approach than we use for rectangular coordinates. Instead of using a grid of x and y to denote a point's location, we describe it using a radial distance and an angle from a central point, usually called the origin.
The radial coordinate, often labeled as \(r\), tells us how far the point is from the origin. The angular coordinate, \(\theta\), indicates the direction or angle from the positive x-axis. This combination of distance and angle is particularly useful in circular or rotational systems, where describing positions relative to a central point simplifies calculations.
For instance, in the given exercise, the polar coordinates \((-4, -\pi / 7)\) mean that the point is 4 units away from the origin in the direction indicated by an angle of \(-\pi / 7\). The negative sign in this context shows direction; a negative radial value reverses the point along its specific angle.

Coordinate Conversion

Converting from polar to rectangular coordinates involves using trigonometric calculations to translate a position defined by distance and angle into one defined by perpendicular coordinate axes. This conversion is necessary when we need to analyze problems within the format of a Cartesian grid.
To accomplish this transformation, we utilize two key formulas. For the x-coordinate:

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

These use the trigonometric functions cosine and sine. The cosine calculates the horizontal distance, while the sine gives the vertical distance. You can plug in the values of \(r\) and \(\theta\) to find the corresponding \(x\) and \(y\) coordinates.
This exercise exemplifies the conversion process using the formulas to transition \((-4, -\pi / 7)\) into \((3.51, 1.90)\) in the rectangular format.

Trigonometric Functions

Trigonometric functions like sine and cosine are integral to the process of converting coordinates. They relate angles in a circle to ratios of side lengths in right triangles. These functions are fundamental in translating angles and distances into positions on a Cartesian plane.
The cosine function measures the horizontal distance from the origin for a given angle, and it ranges from -1 to 1. It is particularly useful in determining the x-coordinate, as seen when computing \(x = r \cos{\theta}\).
On the other hand, the sine function assesses the vertical distance for that angle, instrumental in determining the y-coordinate: \(y = r \sin{\theta}\). The given exercise demonstrated how these trigonometric operations help redefine the position described by polar coordinates into the rectangular system. Understanding these functions and their application is essential for successfully navigating between different coordinate systems.