Chapter 16
The relationship between
trigonometric and
hyperbolic functions
16.1 The relationship between
trigonometric and hyperbolic
functions
In Chapter 21, it is shown that
cos θ + j sin θ = e
j θ
(1)
and cos θ − j sin θ = e
− j θ
(2)
Adding equations (1) and (2) gives:
cos θ =
1
2
(e
jθ
+ e
−jθ )
(3)
Subtracting equation (2) from equation (1) gives:
sin θ =
1
2j
(e
jθ
− e
−jθ )
(4)
Substituting j θ for θ in equations (3) and (4) gives:
cos j θ =
1
2
(e
j ( j θ)
+ e
− j ( j θ)
)
and sin j θ =
1
2 j
(e
j ( j θ)
− e
− j ( j θ)
)
Since j 2 =−1, cos j θ =
1
2 (e −θ + e θ ) =
1
2 (e θ + e −θ )
Hence from Chapter 5, cos j θ = cosh θ
(5)
Similarly, sin j θ =
1
2 j
(e
−θ
− e
θ
) = −
1
2 j
(e
θ
− e
−θ
)
=
−1
j
1
2
(e
θ
− e
−θ
)
= −
1
j
sinh θ (see Chapter 5)
But
−
1
j
= −
1
j
×
j
j
= −
j
j 2 = j,
hence
sin j θ = j sinh θ
(6)
Equations (5) and (6) may be used to verify that in all
standard trigonometric identities, j θ may be written for
θ and the identity still remains true.
Problem 1. Verify that cos 2 j θ + sin 2 j θ = 1.
From equation (5), cos j θ = cosh θ, and from equation (6), sin j θ = j sinh θ.
Thus, cos 2 j θ + sin 2 j θ = cosh 2 θ + j 2 sinh 2 θ, and
since j 2 =−1,
cos
2 j θ + sin
2 j θ = cosh
2
θ − sinh
2
θ