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# 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 ]

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.

### 2 Answers by Expert Tutors

Steve S. | Tutoring in Precalculus, Trig, and Differential CalculusTutoring in Precalculus, Trig, and Diffe...
5.0 5.0 (3 lesson ratings) (3)
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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

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

Hello Steve:
I am thinking of

X + 2X = ∏/ 3
How otherwise can be true , other than

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
Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
0
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.