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Inverse trigonometric functions

Relationships and their inverses

Recall that a relation is a subset of a Cartesian product, that is R⊆A×B or
R : A −→ B while its inverse R−1: B −→ A or R−1= {(y, x) / (x, y) ∈ R}

The graph of R is given by the set of points

{(x, y) / x ∈ DomR; (x, y) ∈ R
and of its inverse relationship
{(y, x) / y ∈ Dom R−1; (x, y) ∈ R}
note that Dom R = Rec R−1∧Dom R−1 = Rec R
grafica 1 ojo



Keeping the variable x, provided to the domain and the variableand for the tour, we have R−1= {(x, y) / x ∈ Dom R−1; (y, x) ∈ R} so Dom R−1 ⊆ eje X ∧ Rec R−1 ⊆ eje Y so graphically

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The graph is obtained:


Dom R = Rec R−1 = [a, b]; a, b ∈ R
Rec R = Dom R−1 = [c, d], c, d ∈ R

Thus the graphs of R and R-1 are symmetrical to eachother with respect to the line bisecting the 1st. quadrant.

Graph of inverse sine Relation
Based on the above we can draw the graph of the inverse of y = sin x (by the symmetry with respect to the line y = x, the graph of the breast)

Grafica 3 ojo gordita hermosa eeee ntc

Immediately, the graphs confirm that it is an inverse relationshipand not a function, then ∀ x ∈ Dom R-1 there areseveral and y ∈R.
notation:

The inverse sine relaci'on is customary in denote by: y = sin -1 x = arcsin x well and in both cases has to be x = sin and.

As −1 ≤ sen y ≤ 1, ∀ y ∈ R =⇒ Dom R−1 = [−1, 1] and so Rec R−1 = R.

But our aim is to discuss the inverse sine function thus restrictingthe travel of the inverse sine relation (or the domain of the sine function), we get "functions" inversesine according to theseintervals (or branches) restricted.

Definitions of inverse trigonometric functions and their graphs

Inverse seno function

We define the inverse sine function in any interval restricted by:

f : [−1, 1] −→ [(2k − 1)π/2, (2k + 1)π/2], k ∈ Z, so

y = f(x) = sen−1x = arcsen x ⇐⇒ x = sen y.

if i k = 0, f : [−1, 1] −→[−π/2,π/2]; is usually called primary rangeand is denoted by:
y = Sen−1x = Arcsen x

Note que Dom f = [−1, 1] y Rec f = [−π/2,π/2]

If k = 0, is usually called, inverse sine in 6 second interval, denoted by y = arcsen x = sen−1x.

We will reverse some breast below a secondary range


k = 1, f : [−1, 1] − = [−π/2,3π/2] , f(x) = sen−1x

k2 = 1, f : [−1, 1] − = [−3π/2,5π/2] , f(x) = sen−1x

observation
Again note that thedomain of any inverse sine function is [-1, 1] which is the route changes

Son 2 Graficas

inverse sine function in its primary range



inverse sine function in one of i ts secondary intervals



Inverse cosine function

Suppose we did the same considerations as for the inversedelseno, ie the graph of its inverse relationship and rectriccionesappropriate and necessary.

So, wedefine the inverse cosine function in any interval, by:

f : [−1, 1] −→ [kπ, (k + 1)π], k ∈ Z, such that
y = f(x) = cos−1x = arccosx ⇐⇒ x = cos y

if k = 0, f : [−1, 1] −→ [0, π], inverse cosine is called in its primary range and is denoted by:

y = Cos−1x = Arccosx
Note that Dom f = [−1, 1] y Rec f = [0, π] ∀k ∈ Z, k = 0 costumba is called, inverse cosine in a secondary range, whichdenotar'a by y = arccosx = cos−1x

Also as for the inverse sine function in any domain for inverse cosine is [-1, 1] and its path is varying.

Dos graficas

inverse cosine function in its primary range (k = 0)

inverse cosine function in one of its secondary intervals (k = -2)


Inverse tangent function

We define the inverse tangent function in any interval resringidoby:

f : R −→((2k −1)(π/2), (2k + 1)(π/2)), k ∈ Z such that y = f(x) = tg −1x = arctg x ⇐⇒ x = tg y

if i k = 0, f : R →(π/2,π/2) such that y = T g−1x = Arctgx is called the inverse tangent in its primary range, n'otese that Dom f = R y su Rec f =(π/2,π/2)

∀k ∈ Z con k = 0 it has what is usually called inverse tangent on one side intervals, denoted by:

y = arctg x = tg−1x

As for the previous case...