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Chapter 8: Problem 31

Write each complex number in rectangular form. 3 cis \(150^{\circ}\)

Short Answer

Expert verified
-\frac{3\sqrt{3}}{2} + i \frac{3}{2}

Step by step solution

01

- Understand the Form

The term 'cis' stands for cosine and sine. The number is in polar form, and can be written as: 3 cis \(150^{\circ}\) = 3 (\cos \(150^{\circ}\) + i \sin \(150^{\circ}\))
02

- Evaluate Cosine and Sine

Find the values of \(\cos 150^{\circ}\) and \(\sin 150^{\circ}\): \(\cos 150^{\circ} = -\frac{\sqrt{3}}{2}\) and \(\sin 150^{\circ} = \frac{1}{2}\)
03

- Substitute and Simplify

Substitute these values into the polar form equation: 3 (\cos \(150^{\circ}\) + i \sin \(150^{\circ}\)) = 3 (-\frac{\sqrt{3}}{2} + i \frac{1}{2}) Simplify this equation to get: = -\frac{3\sqrt{3}}{2} + i \frac{3}{2}

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Form of Complex Numbers
Polar form is a way of expressing complex numbers using magnitudes and angles, making them easier to handle in some mathematical contexts. A complex number in polar form is represented as: \[ r \text{cis} \theta \] where:
  • \(r\) is the magnitude (or modulus) of the complex number.
  • \( \theta \) is the angle (in degrees or radians), also known as the argument.

In our example, the complex number is 3 cis \(150^{\text{\circle}}\). Here, 3 is the magnitude and \(150^{\text{\circle}}\) is the angle.
To convert it to rectangular form, we use trigonometric functions, specifically cosine and sine, to find the real and imaginary parts.
Trigonometric Functions
The trigonometric functions cosine and sine help convert the polar form to rectangular form by providing the real and imaginary components of a complex number.
For our complex number 3 cis \(150^{\text{\circle}}\):
  • The real part is found using \( r \times \text{cos}(\theta)\).
  • The imaginary part is found using \( r \times \text{sin}(\theta)\).

In our example:
  • \( \text{cos}(150^{\text{\circle}}) = -\frac{\text{\sqrt{3}}}{2} \)
  • \( \text{sin}(150^{\text{\circle}}) = \frac{1}{2} \)

By substituting these values, the complex number becomes easier to interpret and use in various calculations.
Complex Number Conversion
Converting between polar and rectangular forms is essential for working with complex numbers in different contexts.
To convert from polar form 3 cis \(150^{\text{\circle}}\) to rectangular form, follow these steps:
Step 1: Use the polar form \( r(\text{cos}(\theta) + i \text{sin}(\theta)) \).
Step 2: Substitute the values of cosine and sine. From above, we have:
  • \( \text{cos}(150^{\text{\circle}}) = -\frac{\text{\sqrt{3}}}{2} \)
  • \( \text{sin}(150^{\text{\circle}}) = \frac{1}{2} \)

Step 3: Simplify the expression:
  • \( 3(\text{cos}(150^{\text{\circle}}) + i \text{sin}(150^{\text{\circle}})) \)
  • Replace \( \text{cos} \) and \( \text{sin} \) values:
  • \( 3(-\frac{\text{\sqrt{3}}}{2} + i \frac{1}{2}) \)
  • Distribute 3:
  • \( -\frac{3\text{\sqrt{3}}}{2} + i \frac{3}{2} \)

The rectangular form of the complex number 3 cis \(150^{\text{\circle}}\) is thus \( -\frac{3\text{\sqrt{3}}}{2} + i \frac{3}{2} \).
This process makes complex mathematics more manageable and opens up various ways to solve complex problems.

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Most popular questions from this chapter

A Heron triangle is a triangle having integer sides and area. Show that each of the following is a Heron triangle. (a) \(a=11, b=13, c=20\) (b) \(a=13, b=14, c=15\) (c) \(a=7, b=15, c=20\) (d) \(a=9, b=10, c=17\)

Electrical Current The current \(I\) in a circuit with voltage \(E,\) resistance \(R,\) capacitive reactance \(X_{c},\) and inductive reactance \(X_{L}\) is $$ I=\frac{E}{R+\left(X_{L}-X_{c}\right) i} $$ Find \(I\) if \(E=12\left(\cos 25^{\circ}+i \sin 25^{\circ}\right), R=3, X_{L}=4,\) and \(X_{c}=6 .\) Give the answer in rectangular form, with real and imaginary parts to the nearest tenth.

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Find the polar equation of the line that passes through the points \(\left(1,0^{\circ}\right)\) and \(\left(2,90^{\circ}\right)\).

Give a complete graph of each polar equation. Also identify the type of polar graph. $$r=3+\cos \theta$$
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