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

Problem

sin^2(x)=3/4

Solution

1. Take the square root of both sides of the equation to isolate the sine function, remembering to include both the positive and negative roots.

sin(x)=±√(,3/4)

2. Simplify the radical on the right side of the equation.

sin(x)=±√(,3)/2

3. Identify the reference angle in the first quadrant where the sine value is √(,3)/2

x=π/3

4. Determine all solutions within one period [0,2*π) by finding the angles in all four quadrants where the sine is either √(,3)/2 or −√(,3)/2

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

5. Write the general solution by adding multiples of the period π since the solutions are spaced exactly π apart.

x=π/3+n*π

x=(2*π)/3+n*π

Final Answer

sin^2(x)=3/4⇒x=π/3+n*π,(2*π)/3+n*π

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