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Chapter 3: Problem 2

Stellen Sie die folgenden Kosinusschwingungen als komplexe Sinuszeiger dar: a) \(y_{1}=3 \cdot \cos (\omega t)\) b) \(y_{2}=-4 \cdot \cos \left(2 t-\frac{\pi}{4}\right)\) c) \(y_{3}=5 \cdot \cos (t+1)\) d) \(y_{4}=3 \cdot \cos \left(\pi t+\frac{2 \pi}{3}\right)\) Anleitung: Bringen Sie die Kosinusschwingungen zunãchst in die Sinusform.

Short Answer

Expert verified
The complex representations are: \(Y_1=3\text{e}^{i(\text{ω}t + \frac{\text{π}}{2})}\), \(Y_2=-4\text{e}^{i(2t + \frac{3\text{π}}{4})}\), \(Y_3=5\text{e}^{i(t + 1 + \frac{\text{π}}{2})}\), \(Y_4=3\text{e}^{i(\text{π}t + \frac{7\text{π}}{6})}\).

Step by step solution

01

- Understand the Cosine to Sine Conversion

Starting with the conversion of a cosine function to a sine function, recall that \(\text{cos}(x) = \text{sin}(x + \frac{\text{π}}{2})\). This will help us in rewriting the equations from cosine to sine.
02

- Convert Equation \(y_1\)

Convert \(y_1=3 \cdot \cos (\text{ω}t)\) to its sine form: \(y_1=3 \cdot \sin(\text{ω}t + \frac{\text{π}}{2}).\) The corresponding complex representation is \(Y_1=3\text{e}^{i(\text{ω}t + \frac{\text{π}}{2})}.\)
03

- Convert Equation \(y_2\)

Convert \(y_2=-4 \cdot \cos (2t - \frac{\text{π}}{4})\) to its sine form: \(y_2=-4 \cdot \sin (2t + \frac{\text{π}}{4} + \frac{\text{π}}{2}).\) Simplify to get \(y_2=-4 \cdot \sin (2t + \frac{3\text{π}}{4}).\) The corresponding complex representation is \(Y_2=-4\text{e}^{i(2t + \frac{3\text{π}}{4})}.\)
04

- Convert Equation \(y_3\)

Convert \(y_3=5 \cdot \cos (t+1)\) to its sine form: \(y_3=5 \cdot \sin (t + 1 + \frac{\text{π}}{2}).\) Simplify to get \(y_3=5 \cdot \sin (t + 1 + \frac{\text{π}}{2}).\) The corresponding complex representation is \(Y_3=5\text{e}^{i(t + 1 + \frac{\text{π}}{2})}.\)
05

- Convert Equation \(y_4\)

Convert \(y_4=3 \cdot \cos (\text{π}t + \frac{2\text{π}}{3})\) to its sine form: \(y_4=3 \cdot \sin(\text{π}t + \frac{2\text{π}}{3} + \frac{\text{π}}{2}).\) Simplify to get \(y_4=3 \cdot \sin(\text{π}t + \frac{7\text{π}}{6}).\) The corresponding complex representation is \(Y_4=3\text{e}^{i(\text{π}t + \frac{7\text{π}}{6})}.\)

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Key Concepts

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

cosine functions
To start with cosine functions, it's important to understand their general form and properties. A cosine function can be expressed as: \( y = A \cos( \text{ω}t + φ) \), where:
  • \(A\) is the amplitude
  • \( \text{ω} \) is the angular frequency
  • \(t\) is the time variable
  • \( φ \) is the phase shift
The cosine function is one of the fundamental functions in trigonometry and plays a key role in describing oscillatory motion.
It starts at its maximum value when \( t = 0 \) if there is no phase shift.
This makes it particularly useful in various fields such as physics and engineering for modeling waves and harmonic motion.
sine functions
Sine functions are another critical part of trigonometry and oscillatory motion. The sine function can be written as: \( y = A \sin(\text{ω}t + φ) \)
The sine function is similar to the cosine function, with the main difference being their phase. In fact, a shifted sine function can represent a cosine function and vice-versa.
This relationship is expressed mathematically as: \( \text{cos}(x) = \text{sin}(x + \frac{π}{2}) \)
This conversion is handy when you need to switch between the two forms.
Sine functions start at zero when \( t = 0 \) (no phase shift) and then oscillates between -A and A.
They are frequently used in signal processing and other technical applications.
complex representation
Complex representation helps extend real-number trigonometry into the complex number field.
When dealing with trigonometric functions, Euler's formula provides a powerful tool.
Euler's formula states: \( e^{ix} = \text{cos}(x) + i \text{sin}(x) \)
This formula allows us to represent a sine or cosine function as a complex exponential.
For example, a cosine function can be written in its complex form as: \( y = A \cos( \text{ω}t + φ) \Rightarrow A e^{i(\text{ω}t + φ)} \)
The complex representation is frequently used in electrical engineering and physics because it simplifies calculations involving oscillatory phenomena.
Euler's formula
Euler's formula provides a stunningly simple connection between trigonometric functions and complex exponentials: \[ e^{ix} = \text{cos}(x) + i \text{sin}(x) \] This equation is derived from the Taylor series expansions of the exponential, cosine, and sine functions.
In addition to its theoretical beauty, Euler's formula has practical applications.
  • It turns trigonometric function multiplications into simpler exponential operations
  • It allows for easy manipulation of oscillatory functions in fields like signal processing, control systems, and quantum mechanics
For our exercise, converting trigonometric functions into their complex forms using Euler's formula simplifies the representation and manipulation of these functions.

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Most popular questions from this chapter

Die in der kartesischen Form gegebenen komplexen Zahlen $$ \begin{array}{llll} z_{1}=2+\pi \mathrm{j} . & z_{2}=4,5-2,4 \mathrm{j}, & z_{3}=-3+5 \mathrm{j}, & z_{4}=-6 \\ z_{5}=-3-2 \mathrm{j}, & z_{6}=-1+\mathrm{j}, & z_{7}=-4 \mathrm{j}, & z_{8}=-3-\mathrm{j} \end{array} $$ sind in die Polarform umzurechnen. Wie lauten die konjugiert komplexen Zahlen?

Mit dem komplexen Zeiger \(\underline{z}=1+2 \mathrm{j}\) werden folgende Operationen durchgeführt: a) \(j \cdot z\) b) \(\quad \underline{z}^{*}\) c) \(\frac{\pi}{j}\) d) \(2.2\) e) \(\mathrm{e}^{\mathrm{J} 30^{\circ}} \cdot z\) f) \(|z|\) g) \(z^{2}\) Stellen Sie diese Operationen in der GauBschen Zahlenebene bildlich dar. Was bedeuten sie geometrisch?

Stellen Sie die folgenden komplexen Zahlen durch Bildpunkte in der GauBschen Zahlenebene symbolisch dar: $$ \begin{array}{lll} z_{1}=3-4 j, & z_{2}=-2+3 j, \quad z_{3}=-5-4 j, & z_{4}=6. \\ z_{5}=3+5 j, & x_{6}=-1-2 j, \quad z_{7}=-4+j, & z_{8}=-3 j \end{array} $$

Zeigen Sie, da \(B\) sich drei gleiche, aber um jeweils \(120^{\circ}\) in der Phase verschobene Wechselspannungen bei der Oberlagerung ausl?schen (die Wechselspannungen entsprechen z B. den drei Phasen eines Drehstroms). Die Rechnung ist im Komplexen durchzuführen.

Berechnen Sie mit den komplexen Zahlen \(z_{1}=-4 \mathrm{j}, \quad z_{2}=3-2 \mathrm{j}, \quad z_{3}=-1+\mathrm{j}\) die folgenden Terme: a) \(z_{1}-2 z_{2}+3 z_{3}\) b) \(\quad 2 z_{1} \cdot z_{2}^{*}\) c) \(\frac{z_{1}^{*} \cdot z_{2}}{z_{3}}\) d) \(z_{1}\left(2 z_{2}^{*}-z_{1}\right)+z_{3}^{*}\) e) \(\frac{z_{1}-z_{2}^{*}}{3 z_{3}^{*}}\) f) \(\frac{z_{1}+z_{3}^{*}}{z_{2}^{*} \cdot z_{3}}\)
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