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Trigonometric to Bijective Function Reduction

Consider the function f : R → [–1, 1] defined by f(x) = sin x, for all x ∈ R.
The function 'f' is a surjection but not an injection on R.
This is because f(2nπ + x) = f(x) for all n ∈ Z and x ∈ R.
i.e., for any t ∈ (–1, 1), there are infinite x ∈ R such that f(x) = t.
But, for any t ∈ (–1, 1), there exists unique x ∈ (–, ) such that f(x) = t.
(–, ) is a sub-interval of a much larger length on which sine function is a bijection (fig).

⇒ The function g : (–, ) → (–1, 1) defined by g(x) = sin x for all x ∈ (–, ) is a bijection. Therefore, it has an inverse function from (– 1, 1) onto (– , ).
Denoting the function 'g' by 'sin', its inverse is denoted by g– 1 or Sin– 1 or arc sin.
Note that the first letter of an inverse trigonometric function is in the upper case.
The inverse sine function is thus defined as:
Sin–1 : (– 1, 1) → (–, )
defined by Sin– 1 x = θ ⇔ θ ∈ (– , )
and sin θ = x

It may be noted that:
(i) if x ∈ (– 1, 0), then Sin– 1 x ∈ (– , 0)
(ii) if x ∈ (0, 1), then Sin– 1 x ∈ (0, )
(iii) Sin– 1 0 = 0

The inverse trigonometric functions of the other ratios i.e., cos, tan, cot, sec and cosec can be defined by taking the suitable domain.

The domains and ranges of the 6 inverse trigonometric functions are summerised in the adjacent table.

Properties of Inverse Trigonometric Functions

1. Reciprocal relation:

  • Sin–1 x = Cosec–1 (1/x) for x ∈ [–1, 0) ∪ (0, 1]
  • Cos–1 x = Sec–1 (1/x) for x ∈ [–1, 0) ∪ (0, 1]
  • Tan–1 x = Cot–1 (1/x) for x > 0
  • Tan–1 x = Cot–1 (1/x) – π for x < 0

2. Same trigonometric functions:

  • Sin–1 (sin θ) = θ for θ ∈ [–, ] and sin (Sin–1 x) = x for x ∈ [–1, 1]
  • Cos–1 (cos θ) = θ for θ ∈ [0, π] and cos (Cos–1 x) = x for x ∈ [–1, 1]
  • Tan–1 (tan θ) = θ for θ ∈ (–, ) and tan (Tan–1 x) = x for x ∈ R
  • Cot–1 (cot θ) = θ for θ ∈ (0, π) an cot (Cot–1 x) = x for x ∈ R
  • Sec–1 (sec θ) = θ for θ ∈ [0, ) ∪ (, π] and sec (Sec–1 x) = x for x ∈ (–∞, –1] ∪ [1, ∞)
  • Cosec–1 (cosec θ) = θ for θ ∈ [–, 0) ∪ (0, ] and
    cosec (Cosec–1 x) = x for x ∈ (–∞, –1] ∪ [1, ∞)
Properties of Inverse Trigonometric Functions ..

3. Formulae for f(–x):

  • Sin–1 (–x) = –Sin– 1 x for x ∈ [–1, 1]
  • Cos–1 (–x) = π – Cos–1 x for x ∈ [–1, 1]
  • Tan–1 (–x) = –Tan–1 x for x ∈ R
  • Cot–1 (–x) = π – Cot–1 x for x ∈ R
  • Sec–1 (–x) = π – Sec–1 x for x ∈ (–∞, –1] ∪ [1, ∞)
  • Cosec–1 (–x) = –Cosec–1 x for x ∈ (–∞, –1] ∪ [1, ∞)

4. Co-trigonometric functions:

  • Sin–1 (cos θ) = – θ for θ ∈ [0,π]
  • Cos–1 (sin θ) = – θ for θ ∈ [–, ]
  • Tan–1 (cot θ) = – θ for θ ∈ (0, π)
  • Cot–1 (tan θ) = – θ for θ ∈ (– , )
  • Sec–1 (cosec θ) = – θ for θ ∈ [– , 0) ∪ (0, ]
  • Cosec–1 (sec θ) = – θ for θ ∈ [0, ) ∪ (, π]
Properties of Inverse Trigonometric Functions ...

5. Inter relations:

  • Sin–1 x = Cos–1 for 0 ≤ x ≤ 1 and Sin–1 x = –Cos–1 for –1 < x ≤ 0
  • Sin–1 x = Tan–1 for x ∈ (–1, 1)
  • Cos–1 x = Sin–1 √(1 – x2) for x ∈ [0, 1] and Cos–1 x = π – Sin–1 √(1 – x2) for x ∈ [–1, 0)
  • Tan–1 x = Sin–1 = Cos–1 for x > 0
  • Cos–1 x + Sin–1 x = for all x ∈ [–1, 1]
  • Tan–1 x + Cot–1 x = for all x ∈ R
  • Sec–1 x + Cosec–1 x = for all x ∈ (–∞, –1] ∪ [1, ∞)

6. f(x) and f(y) relation:

  • Sin–1 x + Sin–1 y = Sin–1 {x + y} for x ≥ 0, y ≥ 0 and x2 + y2 ≤ 1
  • Cos–1 x + Cos–1 y = Cos–1(xy – ) for –1 ≤ x, y ≤ 1 and x + y ≥ 0
  • Tan–1 x + Tan–1 y = Tan–1 for x >0, y > 0 and xy < 1
Graphs of Inverse Trigonometric Functions
Consider the unit hyperbola x2 – y2 = 1, and point P on it whose co-ordinates are (cosh a, sinh a).
Then the area between OP, the hyperbola and x-axis is a/2.
Hyperbolic Functions

Introduction

We know that x2 + y2 = a2 is the equation of a circle with center at origin (0, 0) and radius 'a'.
If x = a cos θ and y = a sin θ, for θ ∈ R, then
a2 cos2 θ + a2sin2 θ = a2(cos2 θ + sin2 θ) = a2(1) = a2
i.e., the point (a cos θ, a sin θ) lies on the circle.
Now, if x = a and y = b
then , which is the equation of a hyperbola.
i.e., points of the form lie on a hyperbola.
The letter 'e' above is a constant number called Euler's number.

'e' is an irrational number (that has never ending and non-repetitive decimal part)

Given by: e = 2.7118281......
2.712 (approximation)

It is the base of natural logarithms.
In limits, it is given by

Definitions

The hyperbolic sine function is a function f : R → R defined by: f(x) = , for all x ∈ R. It is denoted by sinh x.

∴ sinh x = , for all x ∈ R
Similarly, cosh x = , for all x ∈ R
tanh x = , for all x ∈ R
coth x = , for all x ∈ R – {0}
sech x = , for all x ∈ R
cosech x = , for all x ∈ R – {0}

Note: It is easy to understand and conclude the following:
(i) cosh 0 = 1
(ii) sinh 0 = 0
(iii) cosh (– x) = cosh x
(iv) sinh (– x) = – sinh x
(v) cosh x and sech x are even functions
(vi) sinh x, cosech x, tanh x, coth x are odd functions

Identities

I. cosh2 x – sinh2 x = 1
II. 1 – tanh2 x = sech2 x
III. coth2 x – 1 = cosech2 x

Domain and Range of Hyperbolic Functions
Sl.no. Function y = f(x) Domain (x) Range (y)
(i) y = sinh x R R
(ii) y = cosh x R [1, ∞)
(iii) y = tanh x R (– 1, 1)
(iv) y = coth x R – {0} (– ∞, 1) ⋃ (1, ∞)
(v) y = sech x R (0, 1]
(vi) y = cosech x R – {0} R – {0}
Graphs of sinh and cosh functions
Graphs of tanh and coth functions
Graphs of sech and cosech functions
Inverse Hyperbolic Functions

The function f : R → R defined by f(x) = sinh x for all x ∈ R is a bijection.
Hence the inverse of this function exists and is denoted by sinh– 1.
Note that the first letter of an inverse hyperbolic function is in the lower case.
Thus, if 'x' and 'y' are real numbers, then:

sinh– 1 x = y sinh y = x, for all x ∈ R
Similarly cosh– 1 x = y cosh y = x, for all x ∈ (1, ∞)
tanh– 1 x = y tanh y = x, for all x ∈ (– 1, 1)
coth– 1 x = y coth y = x, for all x ∈ R – (– 1, 1)
sech– 1 x = y sech y = x, for all x ∈ (0, 1)
cosech– 1 x = y cosech y = x, for all x ∈ R – {0}

The domain and range of inverse hyperbolic functions are given in the adjacent table.

Graphs of Inverse Hyperbolic Functions
Properties of Hyperbolic Functions

(i) sinh (x + y) = sinh x cosh y + cosh x sinh y
(ii) sinh (x – y) = sinh x cosh y – cosh x sinh y
(iii) cosh (x + y) = cosh x cosh y + sinh x sinh y
(iv) cosh (x – y) = cosh x cosh y – sinh x sinh y
(v) tanh (x + y) =
(vi) tanh (x – y) =
(vii) coth (x + y) = , if x ≠ – y
(viii) coth (x – y) = , if x ≠ y
(ix) sinh 2x = 2 sinh x cosh x =
(x) cosh 2x = cosh2 x + sinh2 x = 2cosh2 x – 1 = 1 + 2sinh2 x =
(xi) tanh 2x =
(xii) coth 2x = , if x ≠ 0

Inverse Hyperbolic Functions as Logarithmic Functions

Catenary Few common examples (Clockwise) A spider's cob-web; High-power transmission lines; An arch; A bridge
Applications of Hyperbolic Functions

Take a rope or chain (with short links) and fix the two ends. Let it hang under the force of gravity. If will naturally form a hyperbolic cosine curve. The shape is called catenary. It has the appearance of parabola, but is not exactly one.

  • In high-power transmission lines, the length of the cable between two successive poles is important. If the length is too short if may snap due to high tension as it stretches. If it is too long, it sags too much posing a threat (as it carries hundreds of KV at hundreds of amperes). More over, the length of the catenary shape is optimum.
  • On a map using Mercator projection, the latitude(L) of a point and its y-coordinate are related by the inverse hyperbolic tangent function.
    y = tanh– 1(L)
  • The catenary curves are used in the design of bridges, arches(Engineering and Architecture) and in suspension of pipelines(off-shore oil & gas industry).
  • The shape assumed by a soap film between two parallel circular rings is called catenoid. Its cross-section is catenary.
  • The threads of a spider's cob-web form several elastic catenaries(see fig.).
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