Homework Help

Verify if arcsinx + arccosx = pi/2 if -1 < x < 1

user profile pic

lillyan | Student, Undergraduate | eNoter

Posted August 20, 2010 at 12:14 AM via web

dislike 0 like

Verify if arcsinx + arccosx = pi/2 if -1 < x < 1

3 Answers | Add Yours

Top Answer

user profile pic

giorgiana1976 | College Teacher | Valedictorian

Posted August 20, 2010 at 12:18 AM (Answer #1)

dislike 1 like

We'll associate a function f(x) to the expression (arcsin x + arccos x).

If we want to prove that the function is a constant function, we'll have to do the first derivative test. If the first derivative is cancelling, that means that f(x) is a constant function, knowing the fact that a derivative of a constant function is 0.

f'(x) = (arcsin x + arccos x)'

f'(x) = 1/sqrt(1-x^2) - 1/sqrt(1-x^2)

We'll eliminate like terms:

f'(x)=0, so f(x)=constant

To verify if the constant is pi/2, we'll put x = 1:

f(1)=arcsin 1 + arccos 1 = pi/2 + 0=pi/2

user profile pic

neela | High School Teacher | Valedictorian

Posted August 20, 2010 at 12:45 AM (Answer #2)

dislike 1 like

Draw   a right angled triangle : with sides BC = x , AC = 1 and AB sqrt(1-x^2)., with B right angle.

Then  angle  SinA = x/1 =x Or  A =  arc sin x....(1).

 cos C = x/1 . Or  C = Arc cosx ...................(2).

From (1) and (2)  A+C  = arc sinx + arc cosx. But  A+ C = 180 -B deg = 180 - 90 = 90. deg or pi/2 rad.

So arcsinx +arc cosx = pi/2.

 

 

user profile pic

hala718 | High School Teacher | (Level 1) Educator Emeritus

Posted August 20, 2010 at 1:21 AM (Answer #3)

dislike 1 like

arcsin x + arccosx = pi/2  if -1<x<1

Let f(x) = arcsinx + arccosx

If f(x) = 1 , then it is a constant , then f'(x) should be 0:

f'(x) = (arcsinx + arccosx)'

       = 1/sqrt(1-x^2) -1/sqrt(1-x^2) = 0

Then we proved that f(x) is a constant number

==> f(x) = C

Now let us substitutw with any value within the interval given.

f(1) = arcsin1 + arccos1 = pi/2 + 0 = pi/2

==> arcsinx + arccosx = pi/2

 

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes