Multiply. √3/2cis(2π/9)·√6/4cis(3π/2). Which is the answer?
A)2√3/8cis(31π/18)
B)3√2/8cis(π/3)
C)3√2/8cis(31π/18)
D)2√3/8cis(π/3)
√3/2cis(2π/9)·√6/4cis(3π/2)
= (√3/2)(√6/4) cis(2π/9 + 3π/2)
= √18/8 cis (31π/18)
= 3√2/8 cis 31π/18
check my arithmetic
Well, with all these square roots and complex numbers, this equation is getting pretty complex itself! But fear not, dear human, for Clown Bot is here to simplify things with a touch of humor.
First, let's multiply the numbers. We multiply the magnitudes and add the angles:
(√3/2) * (√6/4) = √(3*6) / (2*4) = √18 / 8
cis(2π/9) * cis(3π/2) = cis(2π/9 + 3π/2) = cis(4π/18 + 27π/18) = cis(31π/18)
So, the expression simplifies to:
( √18 / 8 ) * ( cis(31π/18) )
And now we look for the answer that matches our simplified expression!
A) 2√3/8cis(31π/18)
B) 3√2/8cis(π/3)
C) 3√2/8cis(31π/18)
D) 2√3/8cis(π/3)
Hmm, let's put on our mathematically funny glasses and analyze these options. I mean, who doesn't like a good math joke?
Option A) 2√3/8 and 31π/18? That doesn't make me laugh.
Option B) 3√2/8 and π/3? Now we're getting somewhere, but it doesn't fit.
Option C) We have a winner! 3√2/8 and 31π/18. That's our answer!
So, the correct answer is:
C) 3√2/8cis(31π/18)
Hope that brought a little bit of laughter and clarity to your mathematical journey!
To multiply complex numbers in polar form, multiply the moduli and add the arguments. Let's break down the given expression step-by-step:
1. √3/2cis(2π/9) = √3/2 * cis(2π/9)
2. √6/4cis(3π/2) = √6/4 * cis(3π/2)
Now, let's multiply these two expressions:
(√3/2 * cis(2π/9)) * (√6/4 * cis(3π/2))
Step 1: Multiply the moduli (coefficients)
√3/2 * √6/4 = (√3 * √6) / (2 * 4) = √18 / 8 = (√9 * √2) / 8 = 3√2 / 8
Step 2: Add the arguments
cis(2π/9) * cis(3π/2) = cis(2π/9 + 3π/2) = cis(4π/18 + 27π/18) = cis(31π/18)
So, the answer is:
3√2/8 * cis(31π/18)
The correct option is C) 3√2/8cis(31π/18).
To multiply complex numbers in polar form, we need to multiply their magnitudes and add their arguments.
√3/2cis(2π/9) and √6/4cis(3π/2) are given in polar form. Let's find the product.
1. Multiply the magnitudes: (√3/2) * (√6/4) = (√3 * √6) / (2 * 4) = (√18) / 8 = (√9 * √2) / 8 = (3√2) / 8.
2. Add the arguments: (2π/9) + (3π/2) = (2π/9) + (27π/18) = (2π + 27π) / 18 = 29π / 18.
Therefore, the product is (3√2/8) cis(29π/18).
Comparing this result with the given options, none of the options match exactly. However, by equating the angles, we can find the closest option.
The angle 29π/18 is equivalent to 31π/18 - 2π/18. We can write this as (31π/18) + (−2π/18).
So, the correct answer is option A) 2√3/8 cis(31π/18).