Problem 20 Represent in the form $x+t y$ ... [FREE SOLUTION]

Chapter 13: Problem 20

Represent in the form $x+t y$ and graph it in the complex plane. $$12\left(\cos \frac{3}{2} \pi+i \sin \frac{3}{2} \pi\right)$$

Short Answer

Expert verified

The complex number is $ 0 - 12i $, a point on the imaginary axis.

Step by step solution

Understand the Complex Expression

The expression given is in the form of a complex number in polar form: $ 12 \left( \cos \frac{3}{2} \pi + i \sin \frac{3}{2} \pi \right) $. Here, the magnitude (or modulus) is 12, and the angle (or argument) is $ \frac{3}{2} \pi $.

Evaluate the Trigonometric Functions

Evaluate $ \cos \frac{3}{2} \pi $ and $ \sin \frac{3}{2} \pi $. - $ \cos \frac{3}{2} \pi = 0 $ (because $ \frac{3}{2} \pi $ corresponds to 270 degrees, where cosine is zero)- $ \sin \frac{3}{2} \pi = -1 $ (since at 270 degrees, sine is -1)

Convert to Rectangular Form

Substitute the values obtained into the complex expression:\[ 12 (0 + i(-1)) = 12 \cdot 0 + 12 \cdot i(-1) = -12i \]So in rectangular form, the complex number is $ 0 - 12i $. This can be written as $ 0 + (-12)i $, where the real part $ x = 0 $ and the imaginary part $ y = -12 $.

Graph the Complex Number

On the complex plane, graph the point corresponding to the complex number $ 0 - 12i $. The complex plane is a two-dimensional plane where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Since the real part $ x = 0 $, the point lies on the imaginary axis. The imaginary part $ -12 $ means we move 12 units downward along the imaginary axis from the origin.

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Key Concepts

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

Polar Form

Polar form is a way of expressing complex numbers, which is particularly useful when dealing with magnitudes and angles. In polar form, a complex number is represented as:

Magnitude: The distance from the origin to the point in the complex plane. In this case, 12.
Angle (or Argument): The direction angle from the positive real axis. Here, it's $ \frac{3}{2} \pi $ radians, equivalent to 270 degrees.

This means the number lies on the negative imaginary axis. The polar form combines these elements into the expression:

$ r( \cos \theta + i \sin \theta ) $

where $ r $ is the magnitude and $ \theta $ is the angle. This form is compact and provides quick insight into the geometric meaning of a complex number, enhancing understanding when performing multiplication or division between complex numbers.

Rectangular Form

Rectangular form is another way to express complex numbers in terms of their components along the real and imaginary axes. This form represents complex numbers as:

$ a + bi $

where $ a $ is the real part, and $ b $ is the imaginary part. For the example provided, the polar form $ 12(\cos \frac{3}{2} \pi + i \sin \frac{3}{2} \pi) $ was converted to a rectangular form as:\[0 - 12i\]In this expression:

Real part $ a = 0 $
Imaginary part $ b = -12 $

The transition from polar to rectangular involves evaluating the trigonometric functions and multiplying by the magnitude, showing how the angle determines the direction. This form is particularly useful for addition, subtraction, and plotting points on the complex plane.

Complex Plane

The complex plane is an essential tool for visualizing complex numbers, similar to how the Cartesian plane is used for real numbers. It consists of two axes:

Real axis: Horizontal, representing real numbers.
Imaginary axis: Vertical, representing imaginary numbers.

In the complex plane, a complex number $ a + bi $ is represented as a point where:

The real component $ a $ determines the position along the horizontal axis.
The imaginary component $ bi $ determines the position along the vertical axis.

For the complex number $ 0 - 12i $, plotting involves:

Staying at 0 on the real axis.
Moving down 12 units on the imaginary axis.

This visual representation aids in grasping the geometric aspects of complex numbers, such as distance and direction, and provides insight into operations like addition and subtraction as movements on this two-dimensional plane.

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91影视

Represent in the form \(x+t y\) and graph it in the complex plane. $$12\left(\cos \frac{3}{2} \pi+i \sin \frac{3}{2} \pi\right)$$

Short Answer

Step by step solution

Understand the Complex Expression

Evaluate the Trigonometric Functions

Convert to Rectangular Form

Graph the Complex Number

Key Concepts

Polar Form

Rectangular Form

Complex Plane

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