Expert: Steve Holleran - 3/26/2008

Question
QUESTION: Please help me solve:
sin 2x = sin 1/2x

ANSWER: Hi Nadya,

The only reasonable way I see to solve this is graphically.  If you enter "sin 2x - sin .5x" into a grapher and find the zeros, you'll get x = 1.2566, 3.7699, 4.1887 for x between 0 and 2pi.

I tried an algebraic approach yesterday, but it took me hours and gave me a bunch of extraneous solutions--it was REALLY awful, so I didn't think it was the way to go.

If the algebraic work was what you needed, and its still relevant, let me know, and I'll try to send it, but just so you know, its really involved and, I think, counterproductive when a graphical solution is so much easier to see.

Hope this helps out.
Steve

---------- FOLLOW-UP ----------

QUESTION: Thank you so much Steve, but actually, I do require the algebraic way of solving this question. Please send it to me if you can. Thank you for your time,
Nadya.

Answer
Hi Nadya,

OK, here goes:

sin 2x = sin 1/2x

2 sin x cos x = sqrt((1-cos x)/2)

4 sin^2 x cos^2 x = (1-cos x)/2

8 sin^2 x cos^2 x = 1 - cos x

8(1 - cos^2 x)cos^2 x = 1 - cos x

8 cos^2 x - 8 cos^4 x + cos x - 1 = 0

-8 cos^4 x + 8 cos^2 x + cos x - 1 = 0

8 cos^4 x - 8 cos^2 x - cos x + 1 = 0

Synthetic division:

1    |8    0    -8    -1    1
     |
     |     8     8     0    -1
     --------------------------

      8    8     0     -1    0     so cos x = 1 is solution

         x = 0, 2pi, 4pi, 6pi,...
But 2pi, 4pi, etc are extraneous, so only x = 0, 4pi, 8pi, ...

Synthetic division again on the quotient polynomial above:

-1/2 | 8     8    0    -1
     |
     |       -4    -2   1
     ------------------------
       8      4    -2   0

         so cos x = -1/2 is solution

         x = 2pi/3, 4pi/3 and multiples

         but 2pi/3 and multiples are extraneous,

         so only 4pi/3 and multiples.

Quotient : 8 cos^2 x + 4 cos x -2 =0

         4 cos^2 x + 2 cos x - 1 = 0

Quadratic formula:

        cos x = [-2 +/- sqrt(20)]/8 = [-1 +/- sqrt(5)] / 4

and      cos x = (-1 + sqrt(5))/4 = .3090

         x = arccos .3090 = 1.2566 radians

       cos x = (-1 -sqrt(5))/4 = -.8090

         x = arccos -.8090 = 2.5133

These last two solutions check out.

I hope you could follow this okay.

Steve