Plane trigonometry, by S.L. Loney.

286

TRIGONOMETRY.

[Exs. XLIV.]

si sin a + s3a + sin 5a +... + sin (2n - 1) a
4. ------. ---- - ------ - -- - ==tan na.
cos a + cos 3a + cos 5a +... cos (2n - 1) a
7r 37r 57r
5. cos 2[1 + cos 2 + cos 2 +... to n terms.
6. cos a - cos (a+ ) + cos (a + 2) -... to 2n terms.
sin a - sin (a + 3) + sin (a + 23) +... to n terms
cos a - cos (a + i) + cos (a + 23) +... to n terms
~n-4 n-6
=tan {a+ 2 r+jS);.
8. sin 0 + sin - 0 + sin _ 2 0 t+... to n terms.
n-2?n-2
9, cosxs + sin 3x cos 5x + sin 7x... + sin (4n - 1) x.
10. sin a sin 2a + sin 2a sin 3a + sin 3a sin 4a +... to n terms.
11. cos a sin 2a + sin 2a cos 3a + cos 3a sin 4a
+ sin 4a cos 5a +... to 2n terms.
12. sin a sin 3a + sin 2a sin 4a + sin 3a sin 5a +... to n terms.
13. cos a cos 3 + cos 3a cos 2p + cos 5a cos 33... to n terms.
14. sin2 a + sin2 2a + sin2 3a +... to n terms.
15. sin2 0 + sin2 (0 + a) + sin2 (0 + 2a) +... to n terms.
16. sin a + sin3 2a + sin3 3a... to n terms.
17. sin4 a + sin4 2a + sin4 3a +... to n terms.
18. cos4 a + cos4 2a - cos4 3a +... to n terms.
19. cos 0 cos 20 cos 30 + cos 20 cos 30 cos 40 +... to n terms.
20. sin a sin (a + 3) - sin (a + /) sin (a + 23) +... to 2n terms.
21. From the sum of the series
sin a + sin 2a + sin 3a +... to n terms,
deduce (by making a very small) the sum of the series
1+2+3+...+n.
22. From the result of the example of Art. 241 deduce the sum of
1+3+5... to n terms.
2r
23. If =
prove that 2 (cos a + cos 2a + cos 4a + cos 5a)
and 2 (cos 3a + cos 5a + cos 6a + cos 7a)
are the roots of the equation
x2 + -4=0.