Further Complex (Trig)

Question 1

How to find sin nx in sinⁿx form

Answer 1

1 - Use Binomial Expansion to expand (cos x + sin x)ⁿ
2 - Use De Moivre’s theorem to expand (cos x + sin x)ⁿ to get (cos nx + sin nx)
3 - Equate the imaginary part of De Moivre’s expression to imaginary part of Binomial expression

Question 2

How to find cos nx in cosⁿx form

Answer 2

1 - Use Binomial Expansion to expand (cos x + sin x)ⁿ
2 - Use De Moivre’s theorem to expand (cos x + sin x)ⁿ to get (cos nx + sin nx)
3 - Equate the real part of De Moivre’s expression to real part of Binomial expression

Question 3

How to find tan nx in tanⁿx form

Answer 3

1 - Use Binomial Expansion to expand (cos x + sin x)ⁿ then divide top and bottom by cosⁿx
2 - Use De Moivre’s theorem to expand (cos x + sin x)ⁿ to get (cos nx + sin nx)
3 - Equate the imaginary/real part of De Moivre’s expression to imaginary/real part of Binomial expression

Question 4

How to use trig to find solutions to polynomials (if roots give exact solutions): Use this as an example
- cos 4x = 8cos⁴x - 8cos²x + 1
- solve 8c⁴ - 8c² + 1 = 0
- find the exact value of cos 3π / 8

Answer 4

1 - make a substitution of a trig function for x (c = cos x)
2 - solve the trig equation (cos 4x = 0) and polynomial (8c⁴ - 8c² + 1 = 0)
3 - order both roots in ascending order
4 - match the root cos3π/8 to the roots of the polynomial

Question 5

How to use trig to find solutions to polynomials (if roots do not give exact solutions): Use this as an example
- cos 4x = 8cos⁴x - 8cos²x + 1
- solve 8c⁴ - 8c² + 1 = 0
- find the exact value of cos3π/8 x cosπ/8

Answer 5

1 - make a substitution of a trig function for x (c = cos x)
2 - solve the trig equation (cos 4x = 0) and polynomial (8c⁴ - 8c² + 1 = 0)
3 - order both roots in ascending order
4 - use roots of polynomials to find product of roots

Question 6

what is the equation for zⁿ + z⁻ⁿ

Answer 6

2cos nx

Question 7

what is the equation for zⁿ - z⁻ⁿ

Answer 7

2isin nx

Question 8

How to find sinⁿx in the form sin nx

Answer 8

1 - Let (z - z⁻¹)ⁿ = (2isinx)ⁿ using identity
2 - expand (z - z⁻¹)ⁿ using binomial expansion
3 - group and factorise pairs of zⁿ, and substitute (2isin nx) for the pairs
4 - equate binomial expression and identity