Find the phasor representation of an equation

[VinnyCee]

Homework Statement

Find the phasors of the following time functions:

(a) $$v(t)\,=\,3\,cos\left(\omega\,t\,-\,\frac{\pi}{3}\right)$$

(b) $$v(t)\,=\,12\,sin\left(\omega\,t\,+\,\frac{\pi}{4}\right)$$

(c) $$i(x,\,t)\,=\,2\,e^{-3\,x}\,sin\left(\omega\,t\,+\,\frac{\pi}{6}\right)$$

(d) $$i(t)\,=\,-2\,cos\left(\omega\,t\,+\,\frac{3\pi}{4}\right)$$

(e) $$i(t)\,=\,4\,sin\left(\omega\,t\,+\,\frac{\pi}{3}\right)\,+\,3\,cos\left(\omega\,t\,-\,\frac{\pi}{6}\right)$$

Homework Equations

A short list of conversions from a larger table in the book. These are conversions from time domain sinusoidal functions on the left to cosine-reference phasor functions on the right.

$$A\,cos\left(\omega\,t\,+\,\phi_0\right)\,\,\iff\,\,A\,e^{j\,\phi_0}$$

$$A\,sin\left(\omega\,t\,+\,\phi_0\right)\,\,\iff\,\,A\,e^{j\left(\phi_0\,-\,\frac{\pi}{2}\right)}$$

The Attempt at a Solution

(a) $$3\,e^{-\frac{\pi}{3}\,j}$$

(b) $$12\,e^{j\,\left(\frac{\pi}{4}\,-\,\frac{\pi}{2}\right)}\,=\,12\,e^{-\frac{\pi}{4}\,j}$$

(c) $$2\,e^{-3\,x}\,e^{j\,\left(\frac{\pi}{6}\,-\,\frac{\pi}{2}\right)}\,=\,2\,e^{-3\,x}\,e^{-\frac{\pi}{3}\,j}\,=\,2\,e^{-3\,x\,-\,\frac{\pi}{3}\,j}$$

(d) $$-2\,e^{\frac{3\pi}{4}\,j}$$

(e) $$4\,e^{j\,\left(\frac{\pi}{3}\,-\,\frac{\pi}{2}\right)}\,+\,3\,e^{j\,\left(-\frac{\pi}{6}\right)}\,=\,7\,e^{j\,\left(-\frac{\pi}{6}\right)}$$I have a question especially with (d), the answer is given as...

$$-2\,e^{j\,\left(\frac{3\pi}{4}\right)}\,=\,-2\,e^{j\,\left(\frac{\pi}{4}\right)}\,=\,2\,e^{-j\,\left(\frac{\pi}{4}\right)}$$

I don't understand how they did the last two conversions in the given answer! Can someone please explain, and say whether the others are correct as well?

Thanks

 

[Marco_84]

what you have to know is just euler formula: exp(ix)=cos(x)+isin(x).