Simplifying and Converting Complex Numbers to Exponential Form

Understanding the exponential form of complex numbers is essential in various mathematical and engineering applications. In this article, we will walk through the process of simplifying a complex number and converting it to the exponential form. The given complex number is:

z frac{sin theta - i cos theta}{sqrt{3} i}

Step 1: Simplify the Denominator

The first step is to simplify the denominator.

z frac{sin theta - i cos theta sqrt{3} - i}{sqrt{3} i sqrt{3} - i}

We can simplify the denominator by noting that:

sqrt{3} ^2 - (-i) ^2 3 - (-1) 3 1 4

Thus, the denominator simplifies to:

z frac{sin theta - i cos theta}{4}

Step 2: Expand the Numerator

Next, we expand the numerator:

sin theta - i cos theta cdot sqrt{3} - i sin theta cdot sqrt{3} - i sin theta - i cos theta cdot sqrt{3} cos theta

Combining like terms, we get:

sin theta cdot sqrt{3} cos theta - i (sin theta sqrt{3} cos theta)

Thus, we have:

z frac{sin theta sqrt{3} cos theta - i (sin theta sqrt{3} cos theta)}{4}

Step 3: Write in Standard Form

We can now write (z) as:

z frac{1}{4} (sin theta sqrt{3} cos theta) - i frac{1}{4} (sin theta sqrt{3} cos theta)

Let:

x frac{1}{4} (sin theta sqrt{3} cos theta)quad y -frac{1}{4} (sin theta sqrt{3} cos theta)

Step 4: Calculate the Modulus (r)

The modulus (r) is given by:

r sqrt{x ^2 y ^2}

Calculating (x^2) and (y^2):

x ^2 left(frac{1}{4} (sin theta sqrt{3} cos theta)right)^2 frac{1}{16} (sin theta sqrt{3} cos theta) ^2 y ^2 left(-frac{1}{4} (sin theta sqrt{3} cos theta)right)^2 frac{1}{16} (sin theta sqrt{3} cos theta) ^2

Thus:

r frac{1}{4} sqrt{(sin theta sqrt{3} cos theta) ^2 (sin theta sqrt{3} cos theta) ^2}

Step 5: Calculate the Argument (phi)

The argument (phi) is given by:

phi tan^{-1}left(frac{y}{x}right)

Substituting the values of (x) and (y), we get:

phi tan^{-1}left(frac{-frac{1}{4} (sin theta sqrt{3} cos theta)}{frac{1}{4} (sin theta sqrt{3} cos theta)}right) tan^{-1}left(frac{-(sin theta sqrt{3} cos theta)}{(sin theta sqrt{3} cos theta)}right)

Step 6: Final Exponential Form

The final exponential form of (z) is:

z r e^{i phi}

Where (r) and (phi) are computed as above. The final expression will depend on the specific values of (theta).

If you need specific values for (r) and (phi) for certain angles (theta), please provide those angles.