all abou the inverse trigonometri, derivative and integral of it

1. THE INVERSE TRIGONOMETRIC FUNCTIONS
By Group I
AlfiramitaHertanti (1111040151)
AmiraAzzahraYunus (1111040153)
DEPARTEMEN OF MATHEMATIC EDUCATION
FACULTY OF SCIENCE AND MATHEMATIC
STATE UNIVERSITY OF MAKASSAR

2. A. DEFINITION
In mathematics, the inverse trigonometric functions (occasionally called
cyclometric function) are the inverse functions of the trigonometric functions (with
suitably restricted domains). Specifically, they are the inverses of the sine, cosine,
tangent, cotangent, secant, and cosecant functions.
Example The sign tan-1-1.374is employed to signify the angle whose tangent
is -1.3674. And in General
Sin-1 x means the angle whose sine is x
Cos-1 x means the angle whose cosine is x
Three pointsshould be noted.
1) Sin-1 x stand for an angle: thus sin-1 ½ = 30o
2) The “-1” is not an index, but merely a sign to denote inversenotation.
3) (sin x)-1 is not used, because it mean the reciprocal of sin x and this is cosec x.
If a functionfis one-to-one on its domain, then f has aninverse function,
denoted by f−1, such that y=f(x) if and onlyif f−1 (y)= x. The domain of f−1 is the range
of f.
The basicidea is that f-1“undoes” what f does, andvice versa. In otherwords,
f−1 (f(x)) =x for all xin the domain of f, and
f (f−1(y)) =y for all yin the range of f.
They are used to obtain an angle from any of the angle's trigonometric ratios.
Inverse trigonometric functions are widely used inengineering, navigation, physics,
and geometry.

3. B. GRAPHICS
a. Inverse Sine and Inverse Cosine
To define the inverse functions for sine and cosine(sometimes called the
arcsineand arccosine and denoted byy=arcsinxor y = arccosine x),
the
domains of these functions are restricted. The restriction thatisplaced on the
domain values of the cosine function is
(see Figure 7-2). This
restricted function is called Cosine. Note the capital “C” in Cosine
The inverse cosine function is defined as the inverse of the restriced
Cosine Function
. Therefore,
Identities for the cosine and inverse cosine:
The inverse sine function’s development is similar to that cosine. The
restriction that is placed on the domain values of the sine function is
This restricted functioniscalled Sine(seeFigure 7-4). Note the capital
"S" in Sine.

4. The inversesine function(seeFigure 7-5) is defined as the inverse of the
restrictedSine functiony = Sin x,
Therefore,
and
Identities for the sineand inverse sine:

5. Thegraphs of the functions y= Cos x and y = Cos-1 x are reflections of
each otherabout the liney = x. The graphs of the functions y = Sin x andy = Sin-1x
are also reflections of each other about the liney = x (see Figure 7-6).
EXAMPLE 1 : Using Figure 7-7, find the exact value of
Thus,
.
Example 2 : Using Figure 7-8, Find the exact value of
Thus,

6. Other Inverse Trigonometric Functions
To define theinversetangent, the domain ofthe tangent must be restricted to
This restricted function is Called Tangent (See Figure 7-9). Note the capital “T” in
Tangent.
The inverse tangent function (see Figure7-10) is defined as theinverseofthe restricted
Tangent function y = Tan x,
Therefore,

7. Identitiesforthe tangent and inverse tangent:
The inverse tangent, inverse secant, and inverse cosecantfunctions are derived
from the restricted Sine, Cosine, and Tangent functions. The graphs of these
functionsare shown in Figure 7-11.

8. Trigonometriidentities involving inverse cotangent, inverse secant, and inverse
cosecant:

9. EXAMPLE 2. Calculate
(a)
(b)
(c)
(d)
SOLUTION
(a)
(b)
(c)
(d)
EXAMPLE 3. Calculate
(a)
(b)
(c)
(d)
SOLUTION
(a)
(b)
(c)
(d)

10. C. PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTION
Four Useful Identities
Theorem A gives some useful identities. You can recall them by reference to the
triangles in Figure 7.
Theorem A
(i)
(ii)
(iii)
(iv)
EXAMPLE 4 Calculate
SOLUTION Recall the double-angel identity

11. From the inverse Function Theorem (Theorem 6.2B), we conclude that sin-1, cos-1,
tan-1, cot-1, csc-1and sec-1 are differentiable. Our aim is to find formulas for their
derivatives.
Theorem B. Derivatives of Inverse Trigonometric Function
(i)
(ii)
-1 < x < 1
-1 < x < 1
(iii)
(iv)
(v)
(vi)
EXAMPLE 5Find
SOLUTION We use Theorem B(i) and the Chain Rule.
Every differential formula leads to an integration formula, a matter we wiil say much
more about in the next chapter. In Particular,
1.
2.
3.
These integration formulas can be generalized slightly to the following:
1.
2.
3.

12. EXAMPLE 6. Evaluate
SOLUTION
Think Of
. Then
+C
Expression as definite Integral
Integrating the derivative and fixing the value at one point gives an expression for the
inverse trigonometric function as a definite integral :

13. When x equals 1, the integral with limited domains are improper integrals, but still
well-defined.
EXAMPLE 7.
Evaluate
SOLUTION
D. SUMMERY
The Inverse Trigonometric Function are the inverse function of
trigonometric function with suitably restricted domains. They are the
inverses of the sine, cosine, tangent, cotangent, secant, and cosecant
functions.
The inverse cosine function is defined as the inverse of the restriced
Cosine Function
. Therefore,
Theinverse sine function (see Figure 7-5) isdefined as the inverse of the
restricted Sine functiony = Sin x,
Identities for the tangent and inverse tangent:
Trigonometric identities involving inverse cotangent, inverse secant, and
inverse cosecant:

14. From the inverse Function Theorem, we conclude that sin-1, cos-1, tan-1,
cot-1, csc-1and sec-1 are differentiable.

15. REFERENCE
Kay, David. 2001. CliffsQuickReviewTM Trigonometry. Hungry Minds,
Inc : New York.
Corral, Michael. 2009.Trigonometry.GNU Free Documentation License
Purcell, Edwind J. 2007. Calculus Ninth Edition. Pearson Education,Ltd :
London.
http://en.wikipedia.org/wiki/inverse_trigonometric_functions