Expert: Steve Holleran - 4/20/2007

Question
What is the value of x, if:
2cosX= 1 + sin X


Answer
Hi Megha,

To solve this one, you have to make use of an algebraic "trick", because there is no way to deal with the separate sine and cosine functions.

If you square both sides of the equation:

     2 cos x = 1 + sin x

    (2 cos x)^2 = (1 + sin x)^2

     4 cos^2 x  = 1 + 2 sin x + sin^2 x

Now replace cos^2 x with 1 - sin^2 x , using the Pythagorean identity:

    4(1 - sin^2 x) = 1 + 2 sin x + sin^2 x

      4 - 4 sin^2 x = 1 + 2 sin x + sin^2 x

Move the left side terms to the right to set = 0:

      0 = 5 sin^2 x + 2 sin x - 3

Now the right side will factor:

      0 = (5 sin x - 3)(sin x + 1)


So the solutions are where   sin x = 3/5   or   sin x = -1

sin x = 3/5 at x = 36.87 degrees

sin x = -1 at x = 270 degrees.

I believe these are the solutions you want.  

Hope this is what you needed.

Steve Holleran