Express the following in the form z=Re[A e^i(ωt+a)] (a) z=sinωt+cosωt. (b) z=cos(ωt-π/ 3)-cosωt

Express the following in the form z=Re[A e^i(ωt+a)] (a)...

Key Concepts

Phasor Representation

Phasor representation involves expressing sinusoids as rotating vectors (phasors) in the complex plane. This method simplifies the process of adding sinusoids of the same frequency and is particularly useful in fields like electrical engineering and signal analysis. By representing a sine or cosine wave as a complex exponential, the combined amplitude and phase shifts can be easily determined through vector addition.

Euler's Formula

Euler's formula, which states that e^(i?) = cos ? + i sin ?, bridges the gap between exponential functions and trigonometric functions. It is a fundamental tool in expressing sinusoidal functions as the real part of a complex exponential, allowing a unified approach to handling oscillatory functions.

Amplitude-Phase Conversion

Amplitude-phase conversion is the technique of rewriting a linear combination of sine and cosine functions of the same frequency into a single sinusoidal function with a new amplitude and phase shift. This conversion is achieved by determining the resultant amplitude using the Pythagorean sum of the coefficients and identifying the appropriate phase angle, thereby simplifying the analysis of the wave's behavior.

Transcript

00:01 Hello students in this question we have to express the following form in z equals to r e to the power j omega t plus alpha okay so this is the form in which we have to represent and the a it is equals to j it is given as sine omega t plus cos omega t okay so we can write this equation as under root 2 of sine omega t sine pi by 4 plus cos omega t and cos pi by 4 okay so we can write this equation as under root 2 of cos omega t minus pi by 4 so this become the answer for this question okay now next so we can write this equation in the form of this form so this can be converted so we can write here that this z will be equals to re and under root 2 this is the amplitude okay and e to the power j omega t minus pi by 4 so this is the answer for the part a.

01:18 Okay now we can write this is the answer for the part a now for the part b in which we have given z it is equals to cause of omega t minus pi by 3 minus cos omega t so after using technometric identities so this equation can be converted to minus of cos omega t plus pi by 3 okay using trigometric identities so this equation can be converted into this form so amplitude is here minus 1 so so z this is equals to r e and e to the power j omega t plus 4 pi by 3 this minus can be converted into here so this equation can be also written as cost of omega t plus 4 pi by 3 okay so amplitude become here 1 so this is the expression for the part b okay now solving for the next part c in which we have given z it is equal to 2.

02:23 Sine omega t plus three cos omega t so using again technometric identities just z this can be converted into under root 13 of cos omega t minus arc 10 of 2 by 3 okay using a this two can be substituted as a this can be substituted as a sign omega x sine omega xan alpha and this 3 can be substituted as a cos alpha okay and after that we will obtain this value okay so we have obtained this value of the z in this form so we can convert this in the form of given form so this j can be also written as amplitude is here under root 13 so r e and amplitude is under root 13 e to the power j and this complete value that is omega t minus arc 10 of 2 by 3.

03:25 So, this become the answer for the part c.

03:29 Now, for the last part d in which we have z, it is equals to sine omega -t minus 2 cos of omega -t minus pi by 4 plus cos omega -t...

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