Posted by **Kate** on Friday, May 17, 2013 at 5:01pm.

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.

- TRIGONOMETRY -
**Reiny**, Friday, May 17, 2013 at 5:52pm
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

- TRIGONOMETRY -
**Steve**, Friday, May 17, 2013 at 5:57pm
(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.

## Answer This Question

## Related Questions

- TRIGONOMETRY - Can you check my work, please? This is for a trig study guide for...
- Trig Help! - Question: Trying to find cos π/12, if cos π/6 = square ...
- TRIGONOMETRY - Can someone please show me step by step how to calculate the ...
- Math, please help - Which of the following are trigonometric identities? (Can be...
- Math/Trig - How would I evaluate these trig functions without using a calculator...
- Math integrals - What is the indefinite integral of ∫ [sin (π/x)]/ x^...
- Pre-Calculus - Find this quotient. Express the result in rectangular form. 9(...
- Pre Calculus - Find each product or quotient. Express the result in rectangular ...
- TRIGONOMETRY - I don't understand how to convert -3-4i to polar coordinates. I ...
- Trig - find all solutions to the equation √3 csc(2theta)=-2 Would the ...

More Related Questions