Plane trigonometry, by S.L. Loney.

316

TRIGONOMETRY.

Ex. 3. If sin a +sin + sin =cos a + cos + cos =0,
prove that cos 3a + cos 3p + cos 3y = 3 cos (a + + y),
and sin 3a + sin 33 + sin 3y= 3 sin (a + + ).
This is an example of the many trigonometrical identities which are
derived from algebraical identities.
For we know that if a + b + c=0,
then a3 + b3 + c3 = 3abc.
Let a=cos a + i sin a, b =cos + i sin p, and c=cos y + i sin y,
so that we have a+b+c=O..'. (cos a + i sin a)3 + (cos p + i sin P)3 + (cos y + i sin )3
= 3 (cos a + i sin a) (cos p + i sin 3) (cos y + i sin y),
so that, by De Moivre's Theorem,
(cos 3a + cos 3, + cos 3y) + i (sin 3a + sin 3p + sin 37)
=3 cos (a + +y) +3i sin (a + 3+y).
Hence, by equating real and imaginary parts, we have the required
results.
EXAMPLES. XLVII.
Put into the form r (cos 0 + i sin 0) the quantities
1, 1+i. 2. -1-i. 3. -/3+i.
4. 3+4i. 5. 1+,2+i. 6. 2-V3+i.
Simplify
(cos 0 - i sin 8)10 (cos a + i sin a) (cos p + i sin 3)
(cos a + i sin a)12' ' (cos y + i sin y) (cos a + i sin 5)
(cos 20 - i sin 28)7 (cos 30 + i sin 30)-5
(cos 40 + i sin40)12 (cos 50 - i sin 50)-6'
cos - i sin )
6(cos (cos a + i sin a)4
10. 6 ' (sin +i cos )5'
1 os + isin
12. (cos - cos q)+i(sin - sin) }n + {cos - cos - i (sin - sin u)}n