Plane trigonometry, by S.L. Loney.

376

TRIGONOMETRY.

[Exs. LV.]

n(n-. 1)
11. sinhx+nsinh2x+ 1 2 sinh 3x +...... to (n + 1) terms
1.2
=2 coshn - sinh (+ 1) x.
12. sinh3 sina+icosh cosa=icos (a + i).
13. sin 2a + i sinh 2p = 2 sin (a + iP) cos (a- i3).
14. cos (a + i3) +isin (a + i)=e- (cos a + i sin a).
15. If tan y = tan a tanh, and tan z = cot a tanh, then prove that
tan (y + z) = sinh 2P cosec 2a.
16. If u=logtan( + prove that tanh-=tan
2/ rv h a =tn2 2
Separate into their real and imaginary parts the quantities
17. cos (a +i). 18. cot (a +pi).
19. cosec (a + i). 20. sec(a +i).
21. sinh (a+ pi). 22. tanh (a + i).
23. sech(a + i).
u + iv sin u +i sinhv
24. Prove that tan -+ =sin u
2 cosu + cosh v
25. If sin (A + iB) =x + iy, prove that
x2 y2 x2 y2
osh B + sinh2 B sin2A cos2A
26. If sin (O + fi) = cos a + i sin a, prove that cos2 0= 4- sin a.
27. If sin (0 + i) =p (cos a + i sin a), prove that
p2= 2 eosh 20 - cos 20] and tan a = tanh 0 cot 0.
28. If cos (0 + -i) = R (cos a + i sin a), prove that
sin (O - a)
= log sin (O + a)'
29. If tan (O + /i) = tan a +i sec a, prove that e2 = - cot, and that
20 =nr + 2 + a.
30. If tan (0 + k<i) = cos a + i sin a, prove that
0= +, and = logtan (+ -
2 2 4 2