Relationship between inverse and sine function, Mathematics

Assignment Help:

Relationship between the inverse sine function and the sine function

We have the given relationship among the inverse sine function and the sine function.

                   sin (sin -1 x ) = x                                     sin -1 (sin x ) =x

In other terms they are inverses of each other. it means that we can utilizes the fact above to determine the derivative of inverse sine.  Let's start with,

f ( x ) = sin x                                                g ( x )= sin -1 x

Then,

g′ ( x ) = 1/ f ′ ( g ( x )) =         1 /cos (sin -1  x )

It is not an extremely useful formula.  Let's see if we can obtain better formula.  Let's begin by recalling the definition of inverse sine function.

                                          y ( sin -1 ( x )         ⇒ x = sin ( y )

By using the first part of this definition the denominator into the derivative becomes,

                                                 cos (sin -1 x ) = cos ( y )

Now, recall that

cos2  y + sin 2  y =1      ⇒  1769_inverse trig.png

By using this, now the denominator is,

547_inverse trig1.png

Now, utilizes the second part of the definition of the inverse sine function. Then the denominator is,

210_inverse trig 2.png

Putting all this together gives the given derivative.

2082_inverse trig3.png


Related Discussions:- Relationship between inverse and sine function

Arc length with vector functions - three dimensional space, Arc Length with...

Arc Length with Vector Functions In this part we will recast an old formula into terms of vector functions.  We wish to find out the length of a vector function, r → (t) =

Marketing, What are the Input and Output of Marketing

What are the Input and Output of Marketing

Linear independence and dependence, It is not the first time that we've loo...

It is not the first time that we've looked this topic. We also considered linear independence and linear dependence back while we were looking at second order differential equation

What is the ratio of the areas of sectors , What is the ratio of the areas ...

What is the ratio of the areas of sectors I and II ?                               (Ans:4:5) Ans:    Ratio will be 120/360  Π r 2 : 150/360  Π r 2 4/12  : 5/12  =

Decimals, how do you turn a fraction into a decimals

how do you turn a fraction into a decimals

Graphical method for interpolation, Graphical Method Whil...

Graphical Method While drawing the graph on a natural scale, the independent variables are marked along the horizontal line and corresponding dependen

Harmonic mean-arthmetic geometric progression, Harmonic mean It is a m...

Harmonic mean It is a measure of central tendency which is utilized to determine the average increase rates for natural economies. This is defined like the reciprocal of the a

Algebra, (x+15)/y=10 where y=5

(x+15)/y=10 where y=5

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd