Solve for x given the interval of [0,2pi).
cos^2x=2+2sinx
I got (3pi)/2 or 270 degrees..
By the way, does the notation [0,2pi) mean
0<=x<2 pi or 360?
Yes, the square bracket means the value is included in the interval, and ( or ) means excluded.
There are other notations where ( or ) is replaced by a square bracket pointing outwards, for example,
[0,2π) is the same as [0,2π[
To solve the above problem:
cos^2x=2+2sinx
Replace cos²x by (1-sin²x) and move everyting to one side to get:
sin²x+2sinx+1=0
Substitute p=sinx and solve the resulting quadratic equation for p.
Then solve for sin(x)=p for x∈ [0,2π).
To solve the equation cos^2x = 2 + 2sinx for the interval [0, 2π), you can follow these steps:
Step 1: Use the Pythagorean identity to express cos^2x in terms of sinx. Since sin^2x + cos^2x = 1, we can write cos^2x = 1 - sin^2x.
Step 2: Substitute this expression into the equation, giving us 1 - sin^2x = 2 + 2sinx.
Step 3: Rearrange the equation to set it equal to zero: sin^2x + 2sinx - 3 = 0.
Step 4: Factor the quadratic equation: (sinx + 3)(sinx - 1) = 0.
Step 5: Set each factor equal to zero and solve for sinx: sinx + 3 = 0 or sinx - 1 = 0.
For sinx + 3 = 0, we have sinx = -3, which is not possible since the range of sine function is between -1 and 1.
For sinx - 1 = 0, we have sinx = 1. This occurs at x = π/2 or 90 degrees.
So the solution for x in the interval [0, 2π) is x = π/2 or 90 degrees.
Regarding the notation [0,2π), it represents the interval from 0 to 2π (or 0 to 360 degrees) inclusive on the lower bound and exclusive on the upper bound. This means that the interval includes 0 but does not include 2π, so it is equivalent to 0 <= x < 2π. Therefore, the correct solution x = π/2 is within this interval.