Search 85,435 tutors
FIND TUTORS
Ask a question
0 0

Solve for x: sin (inverse) x + sin (inverse) 2x = pi/3

Solve for x: sin (inverse) x + sin (inverse) 2x = pi/3
 
[ sin (inverse) refers to the inverse function of sine. Also written as sin^-1 ]

Comments

This morning I added a spreadsheet check and now my answer is Pending Review!
This will be much more easy to do. 
 
Pass the arcsin(2x) to the rigth.  And then.  Just then aply sin in both sides of the ecuation. It will led you to a simple ecuation. And the answer is √(3/28) . Sorry for my english.  Im from venezuela.  
 
Im doing this  through my phone, it will be very difficult to do it all the way.  But it can be done in just 3 or 4 steps tops.  
 
By the way.  The answer of √(21)/14 is exactly the same value of my answer.  Just to help you all to see the things differently, it can be the best way.  
Tutors, please sign in to answer this question.

2 Answers

arcsin(x) + arcsin(2x) = pi/3

t = arcsin(x)
p = arcsin(2x)

t + p = pi/3

sin(t+p) = sin(pi/3) =
= sin(t)cos(p)+cos(t)sin(p) = sqrt(3)/2

x sqrt(1-4x^2) + 2x sqrt(1-x^2) = sqrt(3)/2

x^2 (1-4x^2) + 4 x^2 sqrt(1-5x^2+4x^4) + 4x^2 (1-x^2) = 3/4

x^2 - 4x^4 + 4x^2 - 4x^4 + 4 x^2 sqrt(1-5x^2+4x^4) = 3/4

5x^2 - 8x^4 + 4 x^2 sqrt(1-5x^2+4x^4) = 3/4

4 x^2 sqrt(1-5x^2+4x^4) = 8x^4 - 5x^2 + 3/4

16 x^4 (1 - 5x^2 + 4x^4) = (8x^4 - 5x^2 + 3/4)(8x^4 - 5x^2 + 3/4)

16 x^4 - 80x^6 + 64x^8 =
8x^4(8x^4 - 5x^2 + 3/4) - 5x^2(8x^4 - 5x^2 + 3/4) + 3/4(8x^4 - 5x^2 + 3/4)

16 x^4 - 80x^6 + 64x^8 =
64x^8 - 40x^6 + 6x^4
- 40x^6 + 25x^4 - 15/4 x^2
+ 6x^4 - 15/4 x^2 + 9/16

0 = 21 x^4 - 15/2 x^2 + 9/16

0 = 7 x^4 - 5/2 x^2 + 3/16

0 = 112 x^4 - 40 x^2 + 3

h = - -40/(2*112) = 20/112 = 10/56 = 5/28

k = 3 - 112(5/28)^2 = 21/7 - 25/7 = -4/7

x^2 = 5/28 +- sqrt(- -4/(7*112)) = 5/28 +- sqrt(1/(7*7*4)) = 5/28 +- 2/28

x^2 = 1/4 or 3/28

x = +- 1/2 or +- sqrt(21)/14

+sqrt(21)/14 ~= 0.32732683535399, which is what GeoGebra indicates:
http://www.wyzant.com/resources/files/262177/sum_of_arcsines

What’s the logic in discarding the other 3 answers?

I suppose they will not check; i.e., are extraneous (we did square a couple of times).

Anyone want to do the checks; I’m tired.
 
====
 
Thanks for checking, Parviz!
 
I also decided to do a spreadsheet check and here it is:
 
     x       asin(x)  asin(2x)   B+C   =pi/3?
  0.5000  0.5236  1.5708  2.0944 FALSE
-0.5000 -0.5236 -1.5708 -2.0944 FALSE
 0.3273  0.3335   0.7137  1.0472 TRUE
-0.3273 -0.3335 -0.7137 -1.0472 FALSE

Comments

Hello Steve:
  I am thinking of
   
     X + 2X = ∏/ 3         
                                               How otherwise can be true , other than
 
         X = ∏/9 = 0.3491  , your answer is 0.3273
 
           testing your answer:
                           Sin-1 ( 0. 3273) + Sin-1( 0.654 ) = 1.0463 rad = 59.95°
 
                            Sin-1(0. 3273) = 19. 09        Sin-1 ( 0.654) = 40.84°                   
Sin -1(  0.349) = 20. 43º
 
 Sin -1( 0.349) + Sin-1( 0.698) =
 
            20.48° + 44.26° = 64.74°
Thanks, Parviz!
 
I decided to use a spreadsheet to do the complete check and here it is:
 
     x        asin(x)  asin(2x)   B + C  =pi/3?
  0.5000  0.5236   1.5708   2.0944  FALSE
-0.5000 -0.5236  -1.5708  -2.0944  FALSE
 0.3273  0.3335    0.7137   1.0472  TRUE
-0.3273 -0.3335  -0.7137  -1.0472  FALSE
A° = Sin-1 X       B° = Sin -1 2X     Cos A = √( 1 - X^2)     Cos B = √( 1 - 4X^2)
 
    Sin ( A + B ) =  Sin A Cos B + Cos B Sin A
 
       Sin ( ∏/ 3 ) = X √( 1 - 4X^2) + 2X √( 1- X^2)
 
            √3 / 2 = X √(1 - 4 X^2 ) + 2X √(1 - X^2)
 
                This equation need Graphic calculator to solve.