For the complex number z = StartFraction 5 StartRoot 3 EndRoot Over 4 EndFraction minus StartFraction 5 Over 4 EndFraction i ,what is the polar form?

z = five-halves (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction)

z = five-fourths (cosine (StartFraction pi Over 6 EndFraction) + I sine (StartFraction pi Over 6 EndFraction)

z = five-halves (cosine (StartFraction 11 pi Over 6 EndFraction) + I sine (StartFraction 11 pi Over 6 EndFraction)

z = five-fourths (cosine (StartFraction 11 pi Over 6 EndFraction) + I sine (StartFraction 11 pi Over 6 EndFraction)

To convert the complex number \( z = \frac{5\sqrt{3}}{4} - \frac{5}{4}i \) into polar form, we need to find the modulus \( r \) and the argument \( \theta \).

### Step 1: Find the modulus \( r \)

The modulus \( r \) is given by:
\[
r = \sqrt{\left(\frac{5\sqrt{3}}{4}\right)^2 + \left(-\frac{5}{4}\right)^2}
\]
\[
r = \sqrt{\frac{75}{16} + \frac{25}{16}}
\]
\[
r = \sqrt{\frac{100}{16}} = \sqrt{6.25} = \frac{5}{2}
\]

### Step 2: Find the argument \( \theta \)

The argument \( \theta \) is calculated using:
\[
\theta = \tan^{-1}\left(\frac{-\frac{5}{4}}{\frac{5\sqrt{3}}{4}}\right)
\]
\[
\theta = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right)
\]
\[
\theta = -\frac{\pi}{6}
\]

Since the complex number is located in the fourth quadrant, we add \( 2\pi \) to the argument to get a positive angle:
\[
\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}
\]

### Step 3: Write the polar form

The polar form of \( z \) is:
\[
z = \frac{5}{2} \left( \cos\left(\frac{11\pi}{6}\right) + i\sin\left(\frac{11\pi}{6}\right) \right)
\]

Thus, the correct answer is:
\[
z = \frac{5}{2} \left( \cos\left(\frac{11\pi}{6}\right) + i\sin\left(\frac{11\pi}{6}\right) \right)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you find the polar form of a complex number given in rectangular form?
2. What is the geometric interpretation of converting a complex number to polar form?
3. How do you convert from polar form back to rectangular form?
4. What are the properties of the modulus and argument of a complex number?
5. How does the quadrant affect the calculation of the argument in polar form?

### Tip:
Remember that when finding the argument of a complex number, the quadrant in which the number lies determines the correct angle adjustment to make (adding \( \pi \), \( 2\pi \), etc.).

For the complex number z = (5√3)/4 - 5/4 i, what is the polar form?
Options:
1. z = (5/2)(cos(π/6) + i sin(π/6))
2. z = (5/4)(cos(π/6) + i sin(π/6))
3. z = (5/2)(cos(11π/6) + i sin(11π/6))
4. z = (5/4)(cos(11π/6) + i sin(11π/6))

Learn how to convert the complex number z = (5√3)/4 - 5/4 i to polar form. This problem explores modulus calculation, argument determination, and applying polar coordinates to represent complex numbers. Suitable for high school students studying complex numbers and trigonometry.

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