Problem 37 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-4, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates of the polar point (-4, pi/2) are (0, -4).

Step 1: Understanding the conversion formulas

The conversion from polar coordinates \((r,\theta)\) to rectangular coordinates \((x,y)\) involves the use of trigonometric functions. The formulas are defined as \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\).

Step 2: Application of the formulas

Replace r with -4 and \(\theta\) with \(\pi/2\) in the formulas \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\). This gives us \(x=-4\cos(\pi/2)\) and \(y=-4\sin(\pi/2)\).

Step 3: Simplification of the equations

The cosine of \(\pi/2\) is 0 and the sine of \(\pi/2\) is 1. Thus, \(x=-4*0\) and \(y=-4*1\), this simplifies to \(x=0\) and \(y=-4\). So, the rectangular coordinates of the given polar point are (0, -4).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are a way to define a point in a plane using two numerical values. These values determine the position of the point along two perpendicular axes: the horizontal axis (x-axis) and the vertical axis (y-axis).

• The first value, known as the x-coordinate, tells us how far a point is from the y-axis.
• The second value, called the y-coordinate, indicates how far the point is from the x-axis.

The combination of these two values defines the position of any point on the plane. For example, the point (2,3) is found by starting at the origin (0,0), moving two units in the direction of the x-axis, and then three units upwards along the y-axis.
Rectangular coordinates are useful because they provide a straightforward way to plot points and visualize shapes and patterns on a graph.

Trigonometric Functions

Trigonometric functions are mathematical functions of an angle and are key to understanding relationships in triangles. The most commonly used trigonometric functions in coordinate conversions are sine and cosine.

• Cosine (\(\cos\)) helps find the horizontal component of an angle's reach. For an angle \(\theta\), the formula is \(\cos(\theta)\), yielding values between -1 and 1.
• Sine (\(\sin\)) provides the vertical component. Similarly, \(\sin(\theta)\) also ranges from -1 to 1, depending on the angle.

These functions are fundamental when switching between polar and rectangular coordinates because they break down a point's position into horizontal and vertical components.
For example, to convert polar coordinates like (\(r=3, \theta= \frac{\pi}{4}\)) to rectangular, you calculate:

• \(x = r \cdot \cos(\theta) = 3 \cdot \cos(\frac{\pi}{4})\)
• \(y = r \cdot \sin(\theta) = 3 \cdot \sin(\frac{\pi}{4})\)

Through these functions, we can explore and understand both dimensional and direction aspects of geometric shapes.

Conversion Formulas

Conversion formulas are essential in mathematics for switching between different types of coordinate systems, particularly between polar and rectangular coordinates. Here's how the conversion works:
Polar coordinates give point locations with a radius and an angle \((r, \theta)\). To find their rectangular coordinates \((x, y)\), we use the formulas:

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

These formulas utilize trigonometric functions to determine how far a point is along each axis.
When given a point like \((-4, \frac{\pi}{2})\), use:

• \(x = -4 \cdot \cos(\frac{\pi}{2}) = -4 \cdot 0 = 0\)
• \(y = -4 \cdot \sin(\frac{\pi}{2}) = -4 \cdot 1 = -4\)

This process results in the rectangular coordinates \((0, -4)\), providing a clear and precise location in the plane based on the converted polar information.