Problem 147 - Calculus Volume 3

Question

Find rectangular coordinates for the given point in polar coordinates. $$ \left(0, \frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (0, 0).

Step 1: Understand Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radius or the direct distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis measured in radians.

Step 2: Write the Conversion Formulas

To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]

Step 3: Identify Given Values

From the polar coordinates \((0, \frac{\pi}{2})\), we have \(r = 0\) and \(\theta = \frac{\pi}{2}\).

Step 4: Calculate x-coordinate

Substitute \(r = 0\) and \(\theta = \frac{\pi}{2}\) into the formula for the x-coordinate:\[ x = 0 \cdot \cos\left(\frac{\pi}{2}\right) = 0 \]Since \(r = 0\), \(x = 0\) regardless of the angle.

Step 5: Calculate y-coordinate

Substitute \(r = 0\) and \(\theta = \frac{\pi}{2}\) into the formula for the y-coordinate:\[ y = 0 \cdot \sin\left(\frac{\pi}{2}\right) = 0 \]Similar to \(x\), since \(r = 0\), \(y = 0\) regardless of the angle.

Step 6: Write Rectangular Coordinates

Combine the results from steps 4 and 5. The rectangular coordinates for the given polar point \((0, \frac{\pi}{2})\) are \((0, 0)\).

Key Concepts

Polar Coordinates

Polar coordinates are a handy way to represent a point in a plane.They consist of two values:

• the radius, \(r\), which indicates the straight-line distance from the origin to the point.
• the angle, \(\theta\), which is the angle measured from the positive x-axis in a counter-clockwise direction.

Polar coordinates are especially useful in situations involving circular or rotational symmetries.
For instance, in radar and navigation, where direction and distance from a known point (the origin) is more practical to define the position of objects.The coordinates are typically expressed in the form \((r, \theta)\).
In our example, \((0, \frac{\pi}{2})\), \(r\) is zero, implying the point is at the origin, regardless of the angle \(\theta\).

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane using two values:

• the \(x\)-coordinate, which represents the horizontal distance from the y-axis.
• the \(y\)-coordinate, which represents the vertical distance from the x-axis.

The coordinates are expressed as \((x, y)\).
They are widely used because of their straightforward interpretation and simplicity in math operations like addition and subtraction of vectors.
In the conversion from polar to rectangular coordinates, understanding the geometry of right-angle triangles formed between the origin, the point, and the axes helps visualize the concept.
For instance, coordinates \((0, 0)\) describe the origin in this system.

Coordinate Conversion

Converting between polar and rectangular coordinates involves using trigonometry.This is because we are essentially changing from circular to linear reference frames, akin to projecting shadows.
When converting polar

• \((r, \theta)\) to rectangular \((x, y)\), the formulas are:

\[x = r \cos(\theta)\] \[y = r \sin(\theta)\]These equations come from the definition of sine and cosine in a right triangle.Since \(r = 0\) in the problem example, both \(x\) and \(y\) will equal zero, aligning the point with the origin regardless of \(\theta\).
In general, these formulas allow us to translate from one representation to another seamlessly, which is essential in fields requiring both types of coordinates like graphics programming and engineering.

Radian Measure

Radian measure is a way of expressing angles using the radius of the circle.One radian is the angle formed when the arc length is equal to the radius of the circle.This provides a natural and direct relationship between the angle and the shape of the circle.
Using radians is often more advantageous in calculus and advanced mathematics than degrees.The conversion between degrees and radians is straightforward:

• \(2\pi\) radians equals \(360\) degrees.
• \(\frac{\pi}{2}\) radians equals \(90\) degrees.

In our example, \(\theta = \frac{\pi}{2}\), aligns perfectly with the y-axis upwards, illustrating the angle's position.
Radian measure simplifies many mathematical formulas, leading to more intuitive and elegant expressions.