8.4 Circular Functions of Complex angles & Hyperbolic Function
We have Eular's formulas, e^{iq}
= cosq + i sin q
® (1)
and e^{iq} = cosq
 i sin q ®
(2)
Þ
e^{iq} + e^{iq}=
2 cos q and e^{iq}
 e^{iq} = 2 i sin q
Whence
For any nonreal quantity z, we have sinz =
cos z =
tan z =
with csc z, sec z, cot z as their respective reciprocals.
HYPERBOLIC FUNCTIONS
From the analogy, we define the new functions known as Hyperbolic
functions
and csc hx, sec hx, cot hx are their reciprocals respectively.
Relation between circular and hyperbolic functions
We have sin q =
\
sin(ix) =
Hence we have
(A) sin (ix) = i sin hx (B) sin
hix = i sin x
cos (ix) = cos hx
cos hix = cos x
tan
(ix) = i tan hx
tan hix = i tan x
Note that  ¥ < sin hx <
¥, 1 £ cos hx £
¥ and 1 £ tan hx £
1
Observe the following table
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