Plane trigonometry, by S.L. Loney.

DE MOIVRE'S THEOREM.

321

which is the same as cos + 1 sin
lO4 i-2 + i-2
and this is the first of the quantities already found. Similarly the values
n =5, n=6, n=7 would only give respectively the remaining three quantities, and so on.
Ex. 2. Find all the values of (- 1)3.
Since Cos7= -, and sin7r=0,
we have (- 1) =(cosi r + \/-lsin r)3
= [cos (2n7r + r) + /J - 1 sin (2n7 + 7r)]3
2n- r+7r / — sin2n7r + r
= cos 3 + / - - sin 3
3 3
Giving n the values 0, 1, and 2, the required values are
r J sin5r 57'
os + /-1sin-, cos - sin r, and cos + -sin-,
1+ /-3 1- ^-3
i.e. 2 +,-l,and 2-J
2 2
EXAMPLES. XLVIII.
Find all the values of
1. I1. 2. ( -1)-2. 3. (-i).
4. (-1)1. 5. (1+ /-1). 6. (1+ -— 3).
7. (1-^J-3)0. 8. (V3+J+ -1)'1. 9. (J3- J/- )0.
10. 16. 11. 323. 12. (1+ ^/:3)10+(1- _- 3)10
13. Simplify cos + isin 2)
and express the results in a form free from trigonometrical expressions.
14. Find the continued product of the four values of
/ 7r
cos3-+i sink.
15. Prove that the roots of the equation xl + llx5- 1=0 are
15-1 2?7r,. 2r7r
'- coS 2 i sin -2.
2 5 o
16. Solve the equation x12- 1=0 and find which of its roots satisfy
the equation X4 + 2 + 1= 0.
L. T. 21