Solve for x 4sin(x)^2=1

Problem

4*sin2(x)=1

Solution

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

sin2(x)=1/4

2. Take the square root of both sides, remembering to include both the positive and negative roots.

sin(x)=±√(,1/4)

sin(x)=±1/2

3. Identify the reference angle where the sine function equals 1/2

x=π/6

4. Determine all solutions within the standard interval [0,2*π) for both sin(x)=1/2 and sin(x)=−1/2

x=π/6,(5*π)/6,(7*π)/6,(11*π)/6

5. Generalize the solution to include all possible rotations by adding multiples of π since the values are separated by π radians.

x=π/6+n*π

x=(5*π)/6+n*π

Final Answer

x=π/6+n*π,(5*π)/6+n*π

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