(1) Introduction: If we take x = a cos θ and y = a sin θ (θ ∊ R), then x2 + y2 = a2. In other words, for any real value of θ, the point (a cos θ, a sin θ) lies on the circle x2 + y2 = a2. For this reason the trigonometric functions are known as circular functions.
(4) Domains and Ranges of Hyperbolic Functions:
The six functions defined above are called "Hyperbolic Functions"
Note:
Hence the function f(x) = sin hx (x ∊ R) is an odd function.
(ii) For any x ∊ R, cos h(-x) = = cos hx.
Hence the function f(x) = cos hx (x ∊ R) is an even function
imilarly the hyperbolic functions
f(x) = tan hx, f(x) = cot hx, f(x) = cosec hx are odd functions. And f(x) = sec hx is an even function
(5). Properties Of Hyperbolic function :
1. For x, y ∊ R
i) sin h(x + y) = sin hx cos hy + cos hx sin hy.
ii) sin h(x - y) = sin hx cos hy - cos hx sin hy.
iii) cos h(x + y) = cos hx cos hy + sin hx sin hy.
iv) cos h(x - y) = cos hx cos hy - sin hx sin hy. For any x ∊ R
v) sin h2x = 2 sin hx cos hx =
vi) cos h2x = cos h2 x + sin h2 x = 2 cos h2 x - 1 = 1 + 2 sin h2 x. For x, y ∊ R
xiii) sin h 3x = 3 sin hx + 4 sin h3 x.
xiv) cos h 3x = 4 cos h3 x - 3 cos hx .
6. Inverse Hyperbolic Functions
Definitions :
i) The function f: R → R defined by f(x) = sin hx for all x ∈ R is a bijection, thus the inverse of this function exists and it is denoted by sin h-1. Thus, if x, y are real numbers then
sin h-1 x = y sin hy = x
ii) The function f: [0, ) [1, ∞) defined by f(x) = cos hx, for all x ∈ [0, ∞)), is a bijection,
7. Domain and Range of Inverse Hyperbolic Functions