Problem 72 - Advanced Engineering Mathematics

Question

Convert the given equation to rectangular coordinates. $$ \phi=\pi / 3 $$

Step-by-Step Solution

Answer

The rectangular equation is \(y = \sqrt{3}x\).

Step 1: Understand Polar and Rectangular Coordinates

Polar coordinates are expressed as \((r, \phi)\), where \(r\) is the radius and \(\phi\) is the angle. Rectangular coordinates are \((x, y)\). We will convert the given angle \(\phi = \frac{\pi}{3}\) to its equivalent in rectangular coordinates.

Step 2: Recognize the Equation in Polar Form

The equation given is \(\phi = \frac{\pi}{3}\). This traces a line that passes through the origin making an angle \(\pi/3\) with the x-axis.

Step 3: Apply Conversion Formula

To convert from polar to rectangular coordinates, we use the formulas: \(x = r \cos \phi\) and \(y = r \sin \phi\). Since the line passes through the origin, we can equate \(\cos \phi\) and \(\sin \phi\) directly.

Step 4: Determine Equivalent Rectangular Equation

We use the known values: \(\cos(\frac{\pi}{3}) = \frac{1}{2}\) and \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\). The line equation can be expressed as \(\frac{y}{x} = \frac{\sin(\frac{\pi}{3})}{\cos(\frac{\pi}{3})}\) which simplifies to \(\frac{y}{x} = \sqrt{3}\).

Step 5: Simplify the Rectangular Equation

Solve for \(y\) to express the rectangular equation: \(y = \sqrt{3}x\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a plane using a radius and an angle. Unlike the usual x and y coordinates, polar coordinates tell you how far a point is from the origin and the angle it forms with a reference direction, typically the positive x-axis.
Here are some essential features of polar coordinates:

• The radius (r) represents how far away the point is from the origin.
• The angle (\(\phi\)) is measured from the positive x-axis, typically in radians.
• A point in polar coordinates is written as \((r, \phi)\).

Polar coordinates are handy in situations where circular motion or angles are involved, such as navigation, physics, and engineering problems. For example, if you know that an object is 5 meters from a central point and is positioned at an angle of \(\frac{\pi}{4}\) radians (45 degrees), you've got its polar coordinates!

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two values to define the position of a point on a flat plane. They are based on perpendicular axes: typically one horizontal (x-axis) and one vertical (y-axis).
Here's what you need to know about rectangular coordinates:

• The x-coordinate tells you how far the point is along the horizontal axis.
• The y-coordinate tells you how far the point is along the vertical axis.
• A point is written in the form \((x, y)\), where both x and y can be positive or negative.

Think of rectangular coordinates like a city grid, where each point is a street address. This method is suitable for regions with straight borders and is widely used in many mathematical problems and graphs.

Angle Conversion

Converting angles between radians and degrees is crucial when working with trigonometry and geometry, because angles can be expressed in different units.
Here's how you can understand angle conversion:

• Radians are the standard unit of angular measurement used in many areas of mathematics. There are \(2\pi\) radians in a full circle.
• Degrees divide a circle into 360 parts. It's a common unit in everyday applications.
• To convert from radians to degrees, use the formula: \[ \text{degrees} = \text{radians} \times \frac{180}{\pi} \]
• To convert from degrees to radians, use the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

For example, if you have \(\pi/3\) radians, you can convert it to degrees by multiplying by \(180/\pi\). This gives 60 degrees.

Trigonometry

Trigonometry deals primarily with the relationships between angles and side lengths in right triangles. It provides tools for converting between different coordinate systems, such as polar and rectangular coordinates.
Here are the key trigonometry concepts you need to know:

• The basic trigonometric functions: sine, cosine, and tangent:
  • Sine (\(\sin\)) of an angle gives the ratio of the length of the side opposite the angle to the hypotenuse.
  • Cosine (\(\cos\)) gives the ratio of the adjacent side to the hypotenuse.
  • Tangent (\(\tan\)) is the ratio of the sine to the cosine, or opposite to adjacent.
• These functions allow you to solve for various properties of a triangle or a point when some information is already known. For example, given the angle \(\pi/3\), you can find the sine and cosine to convert between polar and rectangular coordinates.

Trigonometry is a vital tool for many areas of science and engineering, helping us understand the properties of waves, circles, and oscillatory motion.