Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$3.75\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The complex number graphically represented corresponds to a point that is 3.75 units away from the origin and at an angle of \(\frac{3\pi}{4}\) radians counter-clockwise from the real axis. In standard form, the complex number is approximately \(-2.65 + 2.65i\).

Step 1: Visualize the Complex Number

Plot the complex number on the complex plane. Start from the origin, then move along the real axis \(r = 3.75\) units (since \(r\) represents the magnitude of the complex number). Then move an angle of \(\frac{3\pi}{4}\) radians counter clock-wise (since in the polar form the argument \(\theta\) is the angle made with the positive real axis). This point represents the complex number graphically.

Step 2: Convert Polar Form to Rectangular Form

The next step involves finding the rectangular form of the complex number. In rectangular form, a complex number is represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this case, \(a = r\cos\theta\) and \(b = r\sin\theta\). In our case, \(a = 3.75\cos(\frac{3\pi}{4})\) and \(b = 3.75\sin(\frac{3\pi}{4})\). Calculating these will give us the real and imaginary parts, respectively.

Step 3: Calculation and Final Form

Calculate \(a\) and \(b\) using the cosine and sine of \(\frac{3\pi}{4}\) which are \(-\frac{1}{\sqrt{2}}\) and \(\frac{1}{\sqrt{2}}\) respectively. So, \(a = 3.75 \times -\frac{1}{\sqrt{2}}\) equals roughly to \(-2.65\) and \(b = 3.75 \times \frac{1}{\sqrt{2}}\) equals roughly to \(2.65\). Hence, the rectangular (or standard) form of the complex number is approximately \(-2.65 + 2.65i\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent complex numbers, which are numbers with both real and imaginary parts. In polar coordinates, a complex number is expressed as a combination of a magnitude (or modulus) and an angle (or argument). This representation can be very useful for visualizing and performing operations on complex numbers.
Typically denoted as \( r(\cos \theta + i \sin \theta) \) or \( r \operatorname{cis} \theta \), the polar form highlights the number’s rotation and stretch from the origin. Here:

• \(r\) is the modulus or the distance of the point from the origin in the complex plane.
• \(\theta\) is the argument or the angle from the positive real axis, measured counterclockwise.

Polar coordinates elegantly show the geometric significance of complex numbers. When plotted, the modulus \(r\) gives the length, and the angle \(\theta\) represents the direction from the real axis.

Rectangular Form

The rectangular form of a complex number describes it in terms of its real and imaginary components. You might find this form written as \(a + bi\), where \(a\) represents the real part, and \(b\) is the coefficient of the imaginary part \(i\).
Converting from polar to rectangular involves calculating the actual values of these components using trigonometric functions. Specifically:

• The real part \(a\) is determined by \(a = r \cos \theta\).
• The imaginary part \(b\) is found by \(b = r \sin \theta\).

The rectangular form is particularly convenient for addition and subtraction, because it treats the complex number more like two separate numbers. It is also how complex numbers are typically displayed in mathematical expressions.

Conversion of Complex Numbers

To effectively work with complex numbers, it is often necessary to convert between polar and rectangular forms. Each form has its advantages depending on the mathematical operation you're performing.
For conversion from polar to rectangular form as previously mentioned:

• Find the real component by multiplying the modulus \(r\) with the cosine of the argument \(\theta\).
• Compute the imaginary component by multiplying the modulus \(r\) with the sine of the argument \(\theta\).

For example, for the complex number given in the exercise, \(3.75\left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)\), the steps led to calculating approximately \(-2.65 + 2.65i\) in rectangular form by evaluating the trigonometric functions at the specific angle \(\frac{3\pi}{4}\).
Understanding how to switch between these forms not only helps in solving complex equations but also gives a deeper insight into the behavior of complex numbers in various applications like electronics and physics.