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I got an expression like this: $a*\sin(x)+b*\cos(x)+i*c*\sin(x)+i*d*\cos(x)$. I want to calculate the amplitude and the phase of it. Can someone give me some hints? Thx very much!

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    a, b, c, d are real.? – 2017-02-10
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    Yes, all of them are real. – 2017-02-10
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    $$|z|=\sqrt{\left(a\sin(x)+b\cos(x)\right)^2+\left(c\sin(x)+d\cos(x)\right)^2}$$ – 2017-02-10
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    $$arg(z)=\arctan\dfrac{c\sin(x)+d\cos(x)}{a\sin(x)+b\cos(x)}$$ – 2017-02-10
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    @MyGlasses The argument you stated above is not true for every complex number $\text{z}$....!! It depends on the value of the real and imaginary part. – 2017-02-10
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    @JanEerland of course. I wanted to show they are like these and I believe that they not applicable. – 2017-02-10

1 Answers 1

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Assuming that all the variables are real, we can write:

$$\text{Z}=\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)+\left(\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)\right)i\tag1$$

So, for the amplitude:

$$\mathcal{A}=\left|\text{Z}\right|=\sqrt{\Re^2\left(\text{Z}\right)+\Im^2\left(\text{Z}\right)}=\sqrt{\left(\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)\right)^2+\left(\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)\right)^2}\tag2$$

And for the phase, you have to compute:

$$\mathcal{P}=\arg\left(\text{Z}\right)=\arg\left(\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)+\left(\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)\right)i\right)\tag3$$

But, the phase (or argument) does depend on the complex value of $\text{Z}$, for example:

  1. When $\Re\left(\text{Z}\right)=\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)>0$: $$\mathcal{P}=\arg\left(\text{Z}\right)=\arctan\left(\frac{\Im\left(\text{Z}\right)}{\Re\left(\text{Z}\right)}\right)=\arctan\left(\frac{\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)}{\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)}\right)\tag4$$
  2. When $\Re\left(\text{Z}\right)=\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)=0$ and $\Im\left(\text{Z}\right)=\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)>0$: $$\mathcal{P}=\arg\left(\text{Z}\right)=\frac{\pi}{2}=90^\circ\tag5$$
  3. When $\Re\left(\text{Z}\right)=\text{a}\sin\left(x\right)+\text{b}\cos\left(x\right)=0$ and $\Im\left(\text{Z}\right)=\text{c}\sin\left(x\right)+\text{d}\cos\left(x\right)<0$: $$\mathcal{P}=\arg\left(\text{Z}\right)=-\frac{\pi}{2}=-90^\circ\tag6$$

And so on, for the definition of the complex argument look at Wikipedia.

Notice that for the phase there can be a $2\pi=360^\circ$ turn. So for example:

$$\arg\left(4+6i\right)=\arg\left(4+6i\right)+2\pi\text{k}=\arg\left(4+6i\right)+360^\circ\text{k}\tag7$$

Where $\text{k}\in\mathbb{Z}$