TRIGONOMETRY
posted by Kate .
Can someone please give me a step by step answer on how to find the complex cube roots of 8(cos(4π/5)+isin(4π/5))? I need to know how to do this for my trig final next week.

We will use De Moivre's theorem
8(cos(4π/5) + isin(4π/5)^(1/3)
= 8^(1/3) (cos (1/3)(4π/5) + i sin (1/3)(4π/5) )
= 2(cos (4π/15) + i sin(4π/15) )
for a youtube of De Moivre's theorem check out this:
http://www.youtube.com/watch?v=NrwWv9JHdI4 
(rcisθ)^n = r^n cis(nθ)
cuberoot = 1/3 power, so
∛8(cos(4π/5)+isin(4π/5))
= ∛8 cis((4π/5)/3)
= 2cis(4π/15)
Now, since cisθ = cis(θ+2nπ)
cis 4π/5 = cis 14π/5 = cis 24π/5
(2cis 4π/15)^3 = 8cis 4π/5
(2cis (4π/15 + 2π/3))^3 = (2cis 14π/15)^3 = 8cis 14π/5
(2cis (4π/15 + 4π/3))^3 = (2cis 24π/15)^3 = 8cis 24π/5
So, you need to evaluate sin & cos of 4π/15,14π/15,24π/15
adding another 2π takes us around to 34π/15 = 2π + 4π/15 and we're back where we started.
Respond to this Question
Similar Questions

Math integrals
What is the indefinite integral of ∫ [sin (π/x)]/ x^2] dx ? 
Trig Help!
Question: Trying to find cos π/12, if cos π/6 = square root 3 over 2, how to find cos π/12 using DOUBLE angle formula? 
Math/Trig
How would I evaluate these trig functions without using a calculator? 
TRIGONOMETRY
Can you check my work, please? This is for a trig study guide for my final next week. I'd like to know if I'm doing this right. :) 2. The polar coordinates of a point are given. Find the rectangular coordinates of each point. a. (5,(π 
TRIGONOMETRY
Can someone please show me step by step how to calculate the modulus and the argument for these complex numbers? 
TRIGONOMETRY
I don't understand how to convert 34i to polar coordinates. I need to know how so I am ready for my trig final next year. Can someone please show me step by step how to solve this? 
Math, please help
Which of the following are trigonometric identities? 
Pre Calculus
Find each product or quotient. Express the result in rectangular form. 2(cosπ/6+isinπ/6) X 4(cos2π/3 +i2π/3) 
PreCalculus
Find this quotient. Express the result in rectangular form. 9(cos3π/2 + isin3π/2) รท 3(cosπ/4 + isinπ/4) 
Help needed PreCalc
Please help me understand how to solve this questions. I need step by step solution and answer because I find it difficult to understand when someone on here writes the answer instead of showing step by step work on how he/she got …