Plane trigonometry, by S.L. Loney.

HYPERBOLIC FUNCTIONS.

375

ea+ i + e,-a-pi
We have cosh (a+ 3i) = 2 (Art. 313)
ea. ei e-a. e -1i ea (cos + i sin ) + e- (cos - i sinj) 08
= --- ^ -- -2 --- — ^ — 2 ( Art. 308)
2 = 2
cos p (e+ e-a) +i sin (ea- e - )
=2,B — --- -is -- e)= cos p cosh a + i sin3 sinh a.
Aliter. cosh (a +3i)=cos {(a + i) i} (Art. 313)
=cos {ai-pf
= cos (ai) cos B + sin (ai) sin f3
= cosh a cos 3 + i sinh a sin,3.
EXAMPLES. LV.
Prove that
1. cosh 2x=1 + 2 (sinh x)2= 2 (cosh x)2 - 1.
2. cosh (a + ) = cosh a cosh p + sinh a sinh 3.
3. cosh (a + 3) - cosh (a - p) = 2 sinh a sinh 3.
~~~4.,, tanh a + tanh p
4. tanh(a + p)= lh-h
4,tanh (a + tanh a tanh p'
5. cosh 3x = 4 cosh3 x - 3 cosh x.
6. sinh 3x = 3 sinh x 4 sinh x.
7. sinh (x + y) cosh (x - y) = 1 (sinh 2x + sinh 2y).
8. cosh 2x + cosh 5x + cosh 8x + cosh llx
13x 3x
=4 cosh - cosh 3x cosh -
2 2
9. cosh x + cosh (x + y) + cosh (x + 2y) +...... to n terms
cosh x +( - y sinh y
sinh Y
2
10. sinh x + sinh (x + y) + sinh (x + 2y) +...... to n terms
sinh (x +n- y) sinh -
sinh
2