I have to find an exact solutions on the interval [0,pi) for both of these problems--

1) 2sin2x = sqrtx and
2) 1-2sqrt2 sinxcosx = 0

I know this much:
1) sin2x = (sqrt x)/2
= 2sinxcosx = (sqrt x)/2
What do I need to do next?

2) 1/(2 sqrt2) = sinxcosx
= (sqrt2)/4 = sinxcosx
= 2(sqrt2)/4 = 2sincosx
= (sqrt2)/2 = sin2x
And what do I do next?

Thanks!



2) 1/(2 sqrt2) = sinxcosx
= (sqrt2)/4 = sinxcosx
= 2(sqrt2)/4 = 2sincosx
= (sqrt2)/2 = sin2x
And what do I do next?
sin2x= .707
2X= 45 principle angle, 135 second
solve for x
I didn't check your math above.

1) sin2x=1/2 * sqrt(x) my mind is running a blank on this one. If I can see the solution, I will repost.

I still don't understand- What did you mean by 135 second?

the sin of 45 is the same as 90+45...it is symettrical about the 90 degree point.

For problem 1, you correctly simplified the equation to sin2x = (sqrt x)/2. To find the exact solutions on the interval [0, pi), you can solve for x as follows:

1) Start with the equation sin2x = (sqrt x)/2.
2) Square both sides of the equation to eliminate the square root: sin^2(2x) = (sqrt x/2)^2.
3) Use the identity sin^2(2x) = (1 - cos(4x))/2: (1 - cos(4x))/2 = (sqrt x/2)^2.
4) Simplify: 1 - cos(4x) = x/4.
5) Rearrange the equation: 4 - 4cos(4x) = x.
6) Now, you need to solve this equation. Unfortunately, there isn't a straightforward algebraic method to solve it, so you will need to use numerical methods or graphical approaches. You can use software like Wolfram Alpha or graphing calculators to find the approximate solutions.