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using de'Moivres formula evaluate sin(5theta)/sin(theta) and leave your answer in cos(theta)
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This problem is for those students who know complex numbers and Trigonometry very well and want to rise their level from good to perfect.
I will show you how to solve it for more simple (but still very close) formulation:
Using de'Moivres formula evaluate and leave your answer in terms of cos(a).
Solution
Consider complex number z = cos(a) + i*sin(a).
According to de'Moivre formula,
= = . (1)
By applying Newton's binomial formula to and accounting that = -1, you get
= = . (2)
From (1) and (2), you get for the imaginary part coefficients
sin(3a) = - . (3)
Next step divide both sides of (3) by sin(a). You will get
= . (4)
Your last step is to replace in the RHS of (4) by :
= = .
Answer. Using de'Moivre formula, we get = .
Doing by the same way, you can solve the problem for .
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There is a bunch of my lessons on complex numbers in this site
- Complex numbers and arithmetical operations on them
- Complex plane
- Addition and subtraction of complex numbers in complex plane
- Multiplication and division of complex numbers in complex plane
- Raising a complex number to an integer power (*)
- How to take a root of a complex number
- Solution of the quadratic equation with real coefficients on complex domain
- How to take a square root of a complex number
- Solution of the quadratic equation with complex coefficients on complex domain
- Solved problems on taking roots of complex numbers
- Solved problems on arithmetic operations on complex numbers
- Solved problem on taking square root of complex number
- Miscellaneous problems on complex numbers
- Advanced problem on complex numbers
- A curious example of an equation in complex numbers which HAS NO a solution
On de'Moivre formula, see the lesson marked (*) in the list.
Also, you have this free of charge online textbook in ALGEBRA-II in this site
- ALGEBRA-II - YOUR ONLINE TEXTBOOK.
The referred lessons are the part of this online textbook under the topic "Complex numbers".
Save the link to this textbook together with its description
Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson
into your archive and use when it is needed.