find the standard form of the complex number shown below

9(cos 11pi/6 + isin 11pi/6)

a. 2/9 - 2(squareroot)3/9 i
b. -4(squareroot)3/9 + 4/9 i
c. 9/2 - 9(squareroot)3/2 i
d. 9(squareroot)3/2 - 9/2 i
e. -9(squareroot)3/4 - 9/4 i

just evaluate, even if you don't have a calculator, you should be able to find

cos 11π/6 or cos 330° = cos30° = √3/2
sin 11π/6 = -1/2

9(cos 11pi/6 + i sin 11pi/6)
= 9(√3/2 - 1/2)
= 9√3/2 - 9/2 i

Well, let's start by simplifying the expression inside the parentheses.

cos(11π/6) can be simplified to cos(π/6), which is equal to √3/2.

Similarly, sin(11π/6) simplifies to sin(π/6), which is equal to 1/2.

So, the complex number can be written as:

9(√3/2 + i/2)

Now, to find the standard form, we need to rationalize the denominator. Multiplying the numerator and denominator by 2, we get:

(9√3 + 9i) / 2

So, the standard form of the complex number is:

(9√3)/2 + (9i)/2

However, none of the given options match this form. So, it seems there may be a mistake in the options provided. Please double-check the given options, and let me know if you have any other questions!

To find the standard form of the complex number, we can use Euler's formula, which states that e^(iθ) = cos(θ) + isin(θ).

In this case, we have 9(cos(11π/6) + isin(11π/6)). To convert this to the standard form, we need to simplify it.

First, let's simplify the expression inside the parentheses:

cos(11π/6) = cos(π + π/6) = cos(π)cos(π/6) - sin(π)sin(π/6) = (-1)(√3/2) - 0(1/2) = -√3/2

sin(11π/6) = sin(π + π/6) = sin(π)cos(π/6) + cos(π)sin(π/6) = 0(√3/2) + (-1)(1/2) = -1/2

Now, let's substitute these values back into the expression:

9(cos(11π/6) + isin(11π/6)) = 9(-√3/2 + (-1/2)i)

Simplifying further:

= 9(-√3/2) + 9(-1/2)i
= -9√3/2 - 9/2 i

Therefore, the standard form of the complex number is:

-9√3/2 - 9/2 i

So, the correct option is e. -9√3/4 - 9/4 i.

To find the standard form of a complex number, we need to convert it from trigonometric form to standard form by using the following relationships:

1. cos θ = Re(z) / |z|
2. sin θ = Im(z) / |z|

Where:
- θ is the argument angle
- Re(z) is the real part of the complex number
- Im(z) is the imaginary part of the complex number
- |z| is the magnitude or modulus of the complex number

In this case, we have the complex number:

9(cos(11π/6) + i sin(11π/6))

Using the relationships mentioned above, we can evaluate it step by step:

1. Find the magnitude or modulus |z|:
|z| = √(Re(z)² + Im(z)²)
Here, Re(z) = 9 and Im(z) = 0 (since there is no i in the real part)
Therefore, |z| = √(9² + 0²) = 9

2. Find the argument angle θ:
θ = arctan(Im(z) / Re(z))
Here, Re(z) = 9 and Im(z) = 9 sin(11π/6)
θ = arctan((9 * sin(11π/6)) / 9) = arctan(sin(11π/6))

To find the value of sin(11π/6), we need to use the unit circle:

- π/6 = 30 degrees, therefore 11π/6 = 11 * 30 degrees = 330 degrees.
Looking at the unit circle, we find that sin(330 degrees) = -0.5.

Hence, θ = arctan(-0.5)

Now, we need to find the quadrant in which θ lies:
- Since sin(330 degrees) is negative, θ lies in the fourth quadrant.

Now, let's evaluate the angle in the fourth quadrant:

θ = 330 degrees - 360 degrees = -30 degrees

Now, we have |z| = 9 and θ = -30 degrees.

3. Convert back to the standard form:
The standard form of a complex number is:
z = x + yi

Using the relationships:
x = |z| * cos(θ) = 9 * cos(-30 degrees)
y = |z| * sin(θ) = 9 * sin(-30 degrees)

Now, find the values of cos(-30 degrees) and sin(-30 degrees) using the unit circle:

- cos(-30 degrees) = cos(330 degrees) = cos(7π/6) = -√3/2
- sin(-30 degrees) = -sin(30 degrees) = -sin(π/6) = -1/2

Therefore, x = 9 * (-√3/2) = -9√3/2
and y = 9 * (-1/2) = -9/2

Thus, the standard form of the complex number is:
-9√3/2 - 9/2 i

Comparing this to the options provided, we can see that the correct answer is:
e. -9√3/2 - 9/2 i