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

Question

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 ]

Answer 1

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.

 

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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

Answer 2

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.