Solve for x 4sin(x)^2=3

Problem

4*sin2(x)=3

Solution

1. Isolate the squared trigonometric function by dividing both sides of the equation by 4

sin2(x)=3/4

2. Apply the square root property to both sides, remembering to include both the positive and negative roots.

sin(x)=±√(,3/4)

3. Simplify the radical expression on the right side.

sin(x)=±√(,3)/2

4. Identify the angles within one full rotation, [0,2*π) where the sine value is √(,3)/2 or −√(,3)/2

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

5. Generalize the solution to include all possible rotations by adding multiples of π*n (since the solutions are spaced π apart) or 2*π*n for each specific point.

x=π/3+π*n,(2*π)/3+π*n

Final Answer

x=π/3+π*n,(2*π)/3+π*n

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