Problem 9 - Algebra and Trigonometry

Question

Multiple Choice In a rectangular coordinate system, where does the point with polar coordinates \(\left(1,-\frac{\pi}{2}\right)\) lie? (a) in quadrant IV (b) on the \(y\) -axis (c) in quadrant II (d) on the \(x\) -axis

Step-by-Step Solution

Answer

The point lies on the y-axis.

Step 1: Understand polar coordinates

Polar coordinates represent a point in the plane using a distance from a reference point (the origin, usually denoted by \(r\)) and an angle \(\theta\) from a reference direction (the positive x-axis). The given polar coordinates are \(\left(1,-\frac{\pi}{2}\right)\), where \(r = 1\) and \(\theta = -\frac{\pi}{2}\).

Step 2: Convert polar coordinates to rectangular coordinates

To find the Cartesian (rectangular) coordinates, use the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Substituting \(r = 1\) and \(\theta = -\frac{\pi}{2}\), we get: \(x = 1 \cos\left(-\frac{\pi}{2}\right)\) and \(y = 1 \sin\left(-\frac{\pi}{2}\right)\).

Step 3: Evaluate the trigonometric functions

Evaluate the cosine and sine functions: \( \cos\left(-\frac{\pi}{2}\right) = 0\) and \( \sin\left(-\frac{\pi}{2}\right) = -1\). Thus, the rectangular coordinates are \( x = 1 \cdot 0 = 0 \) and \( y = 1 \cdot (-1) = -1 \).

Step 4: Determine the location

The rectangular coordinates are \( (0, -1) \). Since \( x = 0 \), the point lies on the y-axis.

Key Concepts

rectangular coordinate system

The rectangular coordinate system, also known as the Cartesian coordinate system, is a standard framework for locating points in a plane. It comprises two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
A point in this system is denoted as \( (x, y) \) where 'x' is the horizontal distance from the origin, and 'y' is the vertical distance.
This system is divided into four quadrants:

• Quadrant I: Both x and y are positive
• Quadrant II: x is negative, y is positive
• Quadrant III: Both x and y are negative
• Quadrant IV: x is positive, y is negative

Each quadrant helps to determine the location and orientation of a point.

coordinate conversion

Coordinate conversion involves changing coordinates from one system to another. Here, we convert from polar to rectangular coordinates.
Polar coordinates use a radius (r) and an angle (θ). The radius represents the distance from the origin, while the angle determines the direction from the positive x-axis.
The formulas for converting polar coordinates \( (r, \theta) \) to rectangular coordinates \( (x, y) \) are:

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

This involves using trigonometric functions to find the respective horizontal and vertical distances.

trigonometric functions

Trigonometric functions are essential for coordinating conversions. They link angles to the lengths of sides in right-angled triangles.
The primary trigonometric functions are sine (\text{sin}) and cosine (\text{cos}). For an angle \( \theta \):

• \( \cos(\theta) \) gives the adjacent side's length over the hypotenuse.
• \( \sin(\theta) \) gives the opposite side's length over the hypotenuse.

In coordinate conversion, these functions are used to calculate the x and y coordinates from the radius and angle. For example, for \( \theta = -\frac{\pi}{2} \), \( \cos(-\frac{\pi}{2}) = 0 \) and \( \sin(-\frac{\pi}{2}) = -1 \). Therefore, the rectangular coordinates corresponding to polar coordinates \( (1, -\frac{\pi}{2}) \) are \( (0, -1) \).