Solve for x sin(x)^2-3/4=0

Problem

sin2(x)−3/4=0

Solution

1. Isolate the squared trigonometric term by adding 3/4 to both sides of the equation.

sin2(x)=3/4

2. Take the square root of 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 reference angle in the first quadrant where the sine value is √(,3)/2

x=π/3

5. Determine all angles within one period [0,2*π) that satisfy the equation for both positive and negative values.

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

6. Generalize the solution by adding multiples of the period π since the values are spaced exactly π apart (e.g., π/3 and (4*π)/3.

x=π/3+n*π

x=(2*π)/3+n*π

Final Answer

sin2(x)−3/4=0⇒x=π/3+n*π,(2*π)/3+n*π

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