Problem 30 - Precalculus Mathematics for Calculus

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$(-1,5 \pi / 2)$$

Step-by-Step Solution

Answer

The rectangular coordinates are \((0, -1)\).

Step 1: Understand the Conversion

To convert polar coordinates \((-r, \theta)\) to rectangular coordinates \((x, y)\), we use the formulas:\[x = r \cdot \cos(\theta)\]\[y = r \cdot \sin(\theta)\]Since we have a negative \(r\), we need to add \(\pi\) to \(\theta\) to rotate the point by 180 degrees around the origin.

Step 2: Adjust the Angle for Negative Radius

Given the polar coordinates \((-1, \frac{5\pi}{2})\), we adjust \(\theta\) by adding \(\pi\):\[\theta = \frac{5\pi}{2} + \pi = \frac{5\pi}{2} + \frac{2\pi}{2} = \frac{7\pi}{2}\].

Step 3: Standardize the Angle

Angles can be represented in multiple equivalent ways by adding or subtracting complete circles (\(2\pi\)). We'll standardize \(\theta = \frac{7\pi}{2}\) to an equivalent angle between \(0\) and \(2\pi\):\[\frac{7\pi}{2} - 2\pi = \frac{7\pi}{2} - \frac{4\pi}{2} = \frac{3\pi}{2}\].

Step 4: Calculate X Coordinate

Use the formula:\[x = r \cdot \cos(\theta)\]\[x = 1 \cdot \cos\left(\frac{3\pi}{2}\right)\]\[x = 0\]This is because the cosine of \(\frac{3\pi}{2}\) is 0.

Step 5: Calculate Y Coordinate

Use the formula:\[y = r \cdot \sin(\theta)\]\[y = 1 \cdot \sin\left(\frac{3\pi}{2}\right)\]\[y = -1\]This is because the sine of \(\frac{3\pi}{2}\) is -1.

Step 6: Combine Coordinates

The rectangular coordinates for the point are \((0, -1)\).

Key Concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of expressing the location of a point in a plane using two values: the x-coordinate and the y-coordinate. These values represent the horizontal and vertical distances from a reference point called the origin. By using a grid-based system, rectangular coordinates make it easy to pinpoint any location on a flat surface.

Understanding this concept is important because it forms the foundation for many aspects of mathematics and science. It gives a clear, straightforward representation of points that can be easily manipulated and understood. This is especially useful in areas like computer graphics and engineering where precise locations need to be defined and worked with. To convert between different systems, especially when dealing with curves and rotations, an understanding of other coordinate systems such as polar coordinates is essential.

polar coordinates

Polar coordinates offer an alternative way of representing points in a plane, especially useful for scenarios involving circular motion or radial symmetry. Instead of using a grid layout like rectangular coordinates, polar coordinates use a radius and an angle.

The radius indicates how far the point is from the origin, while the angle, typically measured in radians, shows the direction in which you move from the origin to reach the point. One advantage of polar coordinates is their ability to simplify the description of circular and spiral shapes. However, converting between polar and rectangular coordinates requires understanding the relationship between the distance and direction to x and y coordinates.

• This involves using trigonometric functions to compute the equivalent values when moving from one system to another.
• Polar coordinates are highly effective in fields such as physics where motion around a central point is studied, like in analyzing planetary orbits.

trigonometric functions

Trigonometric functions such as sine and cosine are key tools in converting between polar and rectangular coordinates. These functions relate the angles in polar coordinates to the x and y distances in rectangular coordinates.

The cosine of an angle helps determine the x-coordinate by multiplying it with the radius (r), while the sine of the angle determines the y-coordinate. For instance, when given a polar coordinate \((-r, \theta)\), you would need to adjust the angle to account for any negative signs for accurate conversion.

• By adding or subtracting \(\pi\) to the angle \(\theta\), the correct quadrant is specified without changing the position's magnitude.
• Trigonometric functions ensure that the direction and distance relate properly to the corresponding rectangle positions, allowing seamless transitions between coordinate systems.

These functions are not only pivotal in conversions but also for solving numerous problems in calculus and physics, such as wave motion and oscillations.