Problem 13 - Advanced Engineering Mathematics

Question

In Problems 11-14, write the number given in polar form in the form \(a+i b\). $$ z=6\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) $$

Step-by-Step Solution

Answer

The number in Cartesian form is approximately \(5.5434 + 2.2962i\).

Step 1: Understanding the Polar Form

The given complex number is in polar form: \( z = 6(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}) \). This means the complex number is represented as \( r(\cos\theta + i\sin\theta) \), where \( r = 6 \) and \( \theta = \frac{\pi}{8} \). Our goal is to convert this to Cartesian form, \( a + ib \).

Step 2: Using Trigonometric Values

Find the values of \( \cos \frac{\pi}{8} \) and \( \sin \frac{\pi}{8} \). These are standard trigonometric values. Using a calculator or trigonometric tables:\[\cos \frac{\pi}{8} = \frac{\sqrt{2+\sqrt{2}}}{2} \approx 0.9239\]\[\sin \frac{\pi}{8} = \frac{\sqrt{2-\sqrt{2}}}{2} \approx 0.3827\]

Step 3: Calculating the Real Part

Obtain the real part by multiplying \( r \) by \( \cos \theta \):\[a = 6 \times \cos \frac{\pi}{8} = 6 \times 0.9239 = 5.5434\]

Step 4: Calculating the Imaginary Part

Obtain the imaginary part by multiplying \( r \) by \( \sin \theta \):\[b = 6 \times \sin \frac{\pi}{8} = 6 \times 0.3827 = 2.2962\]

Step 5: Combining Real and Imaginary Parts

Combine the real and imaginary parts to express the complex number in the form \( a + ib \):\[z = 5.5434 + 2.2962i\]

Key Concepts

Polar Form

Understanding the polar form of complex numbers is crucial, especially when dealing with mathematical problems involving angles and magnitudes. The polar form expresses a complex number based on its distance from the origin, known as the *modulus*, and its angle from the positive x-axis, known as the *argument*.
Imagine representing a complex number on the complex plane; the polar form is like plotting it using polar coordinates. This form is typically written as \( r (\cos \theta + i \sin \theta) \), where:

• \( r \) is the modulus or magnitude, which is the distance from the origin to the point.
• \( \theta \) is the argument, representing the angle.
• The combination of these gives you a way to describe any complex number on the plane.

In the given problem, the complex number \( z = 6(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}) \) indicates that its modulus is 6 and its argument is \( \frac{\pi}{8} \). Knowing this helps in converting the complex number into its Cartesian form.

Cartesian Form

The Cartesian form of a complex number, also known as the rectangular form, is expressed as \( a + ib \), where \( a \) is the real part and \( b \) is the imaginary part. This form allows one to easily visualize or perform algebraic operations on complex numbers.
To convert a complex number from polar to Cartesian form, you multiply the modulus \( r \) by the cosine of the angle \( \theta \) to find the real part \( a \), and by the sine of \( \theta \) to find the imaginary part \( b \).

• The process: \( a = r \cos \theta \) and \( b = r \sin \theta \).
• Combine these to get \( a + ib \).

In our exercise, we find \( a = 5.5434 \) and \( b = 2.2962 \), making the Cartesian form \( z = 5.5434 + 2.2962i \). Cartesian form is particularly useful for adding, subtracting, and comparing complex numbers.

Trigonometric Values

Understanding and using trigonometric values is key to converting complex numbers between forms. Trigonometric values for angles plotted on the circle, especially standard angles, are useful for precise mathematical work.
In the given exercise, we need \( \cos \frac{\pi}{8} \) and \( \sin \frac{\pi}{8} \). Using trigonometric tables or a calculator, you obtain:

• \( \cos \frac{\pi}{8} \approx 0.9239 \),
• \( \sin \frac{\pi}{8} \approx 0.3827 \).

By substituting these values into the equations for the real and imaginary parts, you can effectively convert between polar and Cartesian forms. These values ensure accuracy when translating complex numbers, maintaining both direction and distance from the origin.

Real and Imaginary Parts

Decomposing a complex number into its real and imaginary parts is necessary for various applications in mathematics and engineering. These parts allow us to handle complex quantities in a way that separates magnitude and direction.
The real part \( a \) is obtained from the product of the modulus \( r \) and the cosine of the angle \( \theta \), while the imaginary part \( b \) results from multiplying \( r \) with the sine of \( \theta \).

• Real part: \( a = r \cos \theta \)
• Imaginary part: \( b = r \sin \theta \)

In our example, the calculations were:

• \( a = 6 \times 0.9239 = 5.5434 \)
• \( b = 6 \times 0.3827 = 2.2962 \)

Thus, the complex number \( z \) is expressed as \( z = 5.5434 + 2.2962i \). This format is essential for complex arithmetic, including addition, subtraction, multiplication, and division.