Tag: complex-number

5 Quick Tips for Converting Cis Form to Rectangular Form

5 Quick Tips for Converting Cis Form to Rectangular Form

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In the realm of mathematics, the conversion of a complex number from its cis (cosine and sine) form to rectangular form is a fundamental operation. Cis form, expressed as z = r(cos θ + i sin θ), provides valuable information about the number’s magnitude and direction in the complex plane. However, for many applications and calculations, the rectangular form, z = a + bi, offers greater convenience and allows for easier manipulation. This article delves into the process of transforming a complex number from cis form to rectangular form, equipping readers with the knowledge and techniques to perform this conversion efficiently and accurately.

The essence of the conversion lies in exploiting the trigonometric identities that relate the sine and cosine functions to their corresponding coordinates in the complex plane. The real part of the rectangular form, denoted by a, is obtained by multiplying the magnitude r by the cosine of the angle θ. Conversely, the imaginary part, denoted by b, is found by multiplying r by the sine of θ. Mathematically, these relationships can be expressed as a = r cos θ and b = r sin θ. By applying these formulas, we can seamlessly transition from the cis form to the rectangular form, unlocking the potential for further analysis and operations.

This conversion process finds widespread application across various mathematical and engineering disciplines. It enables the calculation of impedance in electrical circuits, the analysis of harmonic motion in physics, and the transformation of signals in digital signal processing. By understanding the intricacies of converting between cis and rectangular forms, individuals can unlock a deeper comprehension of complex numbers and their diverse applications. Moreover, this conversion serves as a cornerstone for exploring advanced topics in complex analysis, such as Cauchy’s integral formula and the theory of residues.

Understanding Cis and Rectangular Forms

In mathematics, complex numbers can be represented in two different forms: cis (cosine-sine) form and rectangular form (also known as Cartesian form). Each form has its own advantages and uses.

Cis Form

Cis form expresses a complex number using the trigonometric functions cosine and sine. It is defined as follows:

Z = r(cos θ + i sin θ)

where:

  • r is the magnitude of the complex number, which is the distance from the origin to the complex number in the complex plane.
  • θ is the angle that the complex number makes with the positive real axis, measured in radians.
  • i is the imaginary unit, which is defined as √(-1).

For example, the complex number 3 + 4i can be expressed in cis form as 5(cos θ + i sin θ), where r = 5 and θ = tan-1(4/3).

Cis form is particularly useful for performing operations involving trigonometric functions, such as multiplication and division of complex numbers.

Converting Cis to Rectangular Form

A complex number in cis form (also known as polar form) is represented as (re^{i\theta}), where (r) is the magnitude (or modulus) and (\theta) is the argument (or angle) in radians. To convert a complex number from cis form to rectangular form, we need to multiply it by (e^{-i\theta}).

Step 1: Setup

Write the complex number in cis form and setup the multiplication:

$$(re^{i\theta})(e^{-i\theta})$$

Magnitude \(r\)
Angle \(\theta\)

Step 2: Expand

Use the Euler’s Formula (e^{i\theta}=\cos\theta+i\sin\theta) to expand the exponential terms:

$$(re^{i\theta})(e^{-i\theta}) = r(\cos\theta + i\sin\theta)(\cos\theta – i\sin\theta)$$

Step 3: Multiply

Multiply the terms in the brackets using the FOIL method:

$$\begin{split} &r[(\cos\theta)^2+(\cos\theta)(i\sin\theta)+(i\sin\theta)(\cos\theta)+(-i^2\sin^2\theta)] \\ &= r[(\cos^2\theta+\sin^2\theta) + i(\cos\theta\sin\theta – \sin\theta\cos\theta) ] \\ &= r(\cos^2\theta+\sin^2\theta) + ir(0) \\ &= r(\cos^2\theta+\sin^2\theta)\end{split}$$

Recall that (\cos^2\theta+\sin^2\theta=1), so we have:

$$re^{i\theta} e^{-i\theta} = r$$

Therefore, the rectangular form of the complex number is simply (r).

Breaking Down the Cis Form

The cis form, also known as the rectangular form, is a mathematical representation of a complex number. Complex numbers are numbers that have both a real and an imaginary component. The cis form of a complex number is written as follows:

“`
z = r(cos θ + i sin θ)
“`

where:

  • z is the complex number
  • r is the magnitude of the complex number
  • θ is the argument of the complex number
  • i is the imaginary unit

The magnitude of a complex number is the distance from the origin in the complex plane to the point representing the complex number. The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the point representing the complex number.

In order to convert a complex number from the cis form to the rectangular form, we need to multiply the cis form by the complex conjugate of the denominator. The complex conjugate of a complex number is found by changing the sign of the imaginary component. For example, the complex conjugate of the complex number z = 3 + 4i is z* = 3 – 4i.

Once we have multiplied the cis form by the complex conjugate of the denominator, we can simplify the result to get the rectangular form of the complex number. For example, to convert the complex number z = 3(cos π/3 + i sin π/3) to rectangular form, we would multiply the cis form by the complex conjugate of the denominator as follows:

“`
z = 3(cos π/3 + i sin π/3) * (cos π/3 – i sin π/3)
“`
“`
= 3(cos^2 π/3 + sin^2 π/3)
“`
“`
= 3(1/2 + √3/2)
“`
“`
= 3/2 + 3√3/2i
“`

Therefore, the rectangular form of the complex number z = 3(cos π/3 + i sin π/3) is 3/2 + 3√3/2i.

Plotting the Rectangular Form on the Complex Plane

Once you have converted a cis form into rectangular form, you can plot the resulting complex number on the complex plane.

Step 1: Identify the Real and Imaginary Parts

The rectangular form of a complex number has the format a + bi, where a is the real part and b is the imaginary part.

Step 2: Locate the Real Part on the Horizontal Axis

The real part of the complex number is plotted on the horizontal axis, also known as the x-axis.

Step 3: Locate the Imaginary Part on the Vertical Axis

The imaginary part of the complex number is plotted on the vertical axis, also known as the y-axis.

Step 4: Draw a Vector from the Origin to the Point (a, b)

Use the real and imaginary parts as the coordinates to locate the point (a, b) on the complex plane. Then, draw a vector from the origin to this point to represent the complex number.

Identifying Real and Imaginary Components

To find the rectangular form of a cis function, it’s crucial to identify its real and imaginary components:

Real Component

  • It represents the distance along the horizontal (x) axis from the origin to the projection of the complex number on the real axis.
  • It is calculated by multiplying the cis function by its conjugate, resulting in a real number.

Imaginary Component

  • It represents the distance along the vertical (y) axis from the origin to the projection of the complex number on the imaginary axis.
  • It is calculated by multiplying the cis function by the imaginary unit i.

Using the Table

The following table summarizes how to find the real and imaginary components of a cis function:

Cis Function Real Component Imaginary Component
cis θ cos θ sin θ

Example

Consider the cis function cis(π/3).

  • Real Component: cos(π/3) = 1/2
  • Imaginary Component: sin(π/3) = √3/2

Simplifying the Rectangular Form

To simplify the rectangular form of a complex number, follow these steps:

  1. Combine like terms: Add or subtract the real parts and imaginary parts separately.
  2. Write the final expression in the standard rectangular form: a + bi, where a is the real part and b is the imaginary part.

Example

Simplify the rectangular form: (3 + 5i) – (2 – 4i)

  1. Combine like terms:
    • Real parts: 3 – 2 = 1
    • Imaginary parts: 5i – (-4i) = 5i + 4i = 9i
  2. Write in standard rectangular form: 1 + 9i

Simplifying the Rectangular Form with a Calculator

If you have a calculator with a complex number mode, you can simplify the rectangular form as follows:

  1. Enter the real part in the real number part of the calculator.
  2. Enter the imaginary part in the imaginary number part of the calculator.
  3. Use the appropriate function (usually “simplify” or “rect”) to simplify the expression.

Example

Use a calculator to simplify the rectangular form: (3 + 5i) – (2 – 4i)

  1. Enter 3 into the real number part.
  2. Enter 5 into the imaginary number part.
  3. Use the “simplify” function.
  4. The calculator will display the simplified form: 1 + 9i.

How to Get a Cis Form into Rectangular Form

To convert a cis form into rectangular form, you can use the following steps:

  1. Multiply the cis form by 1 in the form of $$(cos(0) + isin(0))$$
  2. Use the trigonometric identities $$cos(α+β)=cos(α)cos(β)-sin(α)sin(β)$$ and $$sin(α+β)=cos(α)sin(β)+sin(α)cos(β)$$ to simplify the expression.

Advantages and Applications of Rectangular Form

The rectangular form is advantageous in certain situations, such as:

  • When performing algebraic operations, as it is easier to add, subtract, multiply, and divide complex numbers in rectangular form.
  • When working with complex numbers that represent physical quantities, such as voltage, current, and impedance in electrical engineering.

Applications of Rectangular Form:

The rectangular form finds applications in various fields, including:

Field Application
Electrical Engineering Representing complex impedances and admittances in AC circuits
Signal Processing Analyzing and manipulating signals using complex Fourier transforms
Control Systems Designing and analyzing feedback control systems
Quantum Mechanics Describing the wave function of particles
Finance Modeling financial instruments with complex interest rates

Converting Cis Form into Rectangular Form

To convert a complex number from cis form (polar form) to rectangular form, follow these steps:

  1. Let \(z = r(\cos \theta + i\sin \theta)\), where \(r\) is the modulus and \(\theta\) is the argument of the complex number.
  2. Multiply both sides of the equation by \(r\) to obtain \(rz = r^2(\cos \theta + i\sin \theta)\).
  3. Recognize that \(r^2 = x^2 + y^2\) and \(r(\cos \theta) = x\) and \(r(\sin \theta) = y\).
  4. Substitute these values into the equation to get \(z = x + yi\).

Real-World Examples of Cis Form to Rectangular Form Conversion

Example 1:

Convert \(z = 4(\cos 30° + i\sin 30°)\) into rectangular form.

Using the steps outlined above, we get:

  1. \(r = 4\) and \(\theta = 30°\)
  2. \(x = r\cos \theta = 4 \cos 30° = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3}\)
  3. \(y = r\sin \theta = 4 \sin 30° = 4 \times \frac{1}{2} = 2\)

Therefore, \(z = 2\sqrt{3} + 2i\).

Example 2:

Convert \(z = 5(\cos 120° + i\sin 120°)\) into rectangular form.

Following the same steps:

  1. \(r = 5\) and \(\theta = 120°\)
  2. \(x = r\cos \theta = 5 \cos 120° = 5 \times \left(-\frac{1}{2}\right) = -2.5\)
  3. \(y = r\sin \theta = 5 \sin 120° = 5 \times \frac{\sqrt{3}}{2} = 2.5\sqrt{3}\)

Hence, \(z = -2.5 + 2.5\sqrt{3}i\).

Additional Examples:

Cis Form Rectangular Form
\(10(\cos 45° + i\sin 45°)\) \(10\sqrt{2} + 10\sqrt{2}i\)

\(8(\cos 225° + i\sin 225°)\) \(-8\sqrt{2} – 8\sqrt{2}i\)

\(6(\cos 315° + i\sin 315°)\) \(-3\sqrt{2} + 3\sqrt{2}i\)

Troubleshooting Common Mistakes in Conversion

Mistakes when converting cis to rectangular form:

– Incorrect signs: Make sure you use the correct signs for the real and imaginary parts when converting back from cis form.
– Missing the imaginary unit: When converting from cis to rectangular form, remember to include the imaginary unit i for the imaginary part.
– Confusing radians and degrees: Ensure that you are using radians for the angle in the cis form, or convert it to radians before performing the conversion.
– Errors in trigonometric identities: Use the correct trigonometric identities when calculating the real and imaginary parts, such as sin(a + b) = sin(a)cos(b) + cos(a)sin(b).
– Decimal rounding errors: To avoid inaccuracies, use a calculator or a computer program to perform the conversion to minimize rounding errors.
– Incorrect angle range: The angle in the cis form should be within the range of 0 to 2π. If the angle is outside this range, adjust it accordingly.
– Absolute value errors: Check that you are taking the absolute value of the modulus when converting the complex number back to rectangular form.

Summary of the Conversion Process

Converting a cis form into rectangular form involves two primary steps: converting the cis form into exponential form and then transitioning from exponential to rectangular form. This process allows for a better understanding of the complex number’s magnitude and angle.

To convert a cis form into exponential form, raise the base e (Euler’s number) to the power of the complex exponent, where the exponent is given by the argument of the cis form.

The next step is to convert the exponential form into rectangular form using Euler’s formula: e^(ix) = cos(x) + isin(x). By substituting the argument of the exponential form into Euler’s formula, we can determine the real and imaginary parts of the rectangular form.

Cis Form Exponential Form Rectangular Form
cis(θ) e^(iθ) cos(θ) + isin(θ)

Converting from Exponential to Rectangular Form (Detailed Steps)

1. Determine the angle θ from the exponential form e^(iθ).

2. Calculate the cosine and sine of the angle θ using a calculator or trigonometric table.

3. Substitute the values of cos(θ) and sin(θ) into Euler’s formula:

e^(iθ) = cos(θ) + isin(θ)

4. Extract the real part (cos(θ)) and the imaginary part (isin(θ)).

5. Express the complex number in rectangular form as: a + bi, where ‘a’ is the real part and ‘b’ is the imaginary part.

6. For example, if e^(iπ/3), θ = π/3, then cos(π/3) = 1/2 and sin(π/3) = √3/2. Substituting these values into Euler’s formula gives: e^(iπ/3) = 1/2 + i√3/2.

How To Get A Cis Form Into Rectangular Form

To get a cis form into rectangular form, you need to multiply the cis form by the complex number $e^{i \theta}$, where $\theta$ is the angle of the cis form. This will give you the rectangular form of the complex number.

For example, to get the rectangular form of the cis form $2(\cos 30^\circ + i \sin 30^\circ)$, you would multiply the cis form by $e^{i 30^\circ}$:

$$2(\cos 30^\circ + i \sin 30^\circ) \cdot e^{i 30^\circ} = 2\left(\cos 30^\circ \cos 30^\circ + i \cos 30^\circ \sin 30^\circ + i \sin 30^\circ \cos 30^\circ – \sin 30^\circ \sin 30^\circ\right)$$

$$= 2\left(\cos 60^\circ + i \sin 60^\circ\right) = 2\left(\frac{1}{2} + \frac{i \sqrt{3}}{2}\right) = 1 + i \sqrt{3}$$

Therefore, the rectangular form of the cis form $2(\cos 30^\circ + i \sin 30^\circ)$ is $1 + i \sqrt{3}$.

People Also Ask About How To Get A Cis Form Into Rectangular Form

What is the difference between cis form and rectangular form?

The cis form of a complex number is written in terms of its magnitude and angle, while the rectangular form is written in terms of its real and imaginary parts. The cis form is often used in trigonometry and calculus, while the rectangular form is often used in algebra and geometry.

How do I convert a rectangular form into cis form?

To convert a rectangular form into cis form, you need to use the following formula:

$$a + bi = r(\cos \theta + i \sin \theta)$$

where $a$ and $b$ are the real and imaginary parts of the complex number, $r$ is the magnitude of the complex number, and $\theta$ is the angle of the complex number.