Plane trigonometry, by S.L. Loney.

434

TRIGONOMETRY.

Ex. 2. If n be even, prove that
-1.27r.4r r 6 n-2
2 sin 2t sin 2sin......sin ~ i == %/n.
2n 2? 2nn 2n
In equation (2) of Art. 366 put n equal to unity.
xn _ 1 Xn-l + Xn-2.~...... +x+l
Then, since _1= --- X x+1
xln-1 n
therefore, when x is unity, 2 1 =
Hence we have
2 - 2 cos 2 2- 2cos...... 2 - 2cos 7r
2 n n n
2w 4 n - 2
i.e. n=2.4sin2 -. 4 sin2......4sin 2 n,
there being _- 1 factors,
2w 4w n-2
= 2n-. sin2 sin2...... sin2 r.
2n 2- 2?
nl 2w. 4wr n-2
Hence 4- /n=2 2 sin 2-s sin (...... sin................1)
27r 47r n,- 2
Each of the angles - -- 2r is less than a right angle, so
that each of the sines on the right-hand side of (1) is positive.
On the left-hand side we therefore replace the ambiguity by the
positive sign and have the required result.
EXAMPLES. LXIV.
Factorize the following quantities.
1. x6 + 23cos 120 + 1. 2. x8 2x4 cos 60~+ 1.
3. x1 -2x5cos+l. 4. X12 + +1.
5. x14+X7+1. 6. x5-1. 7. x+1.
8. x7-1. 9. X9+1. 10. x10-1.
11. x13+ 1. 12. x14- 1. 13. 20 1.