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. Apply the square root property to solve for sin(x) remembering to include both the positive and negative roots.

sin(x)=±√(,1/4)

3. Simplify the square root on the right side of the equation.

sin(x)=±1/2

4. Identify the reference angle for which the sine value is 1/2 In the first quadrant, this occurs at π/6 or 30

(x_ref)=π/6

5. Determine all solutions within one period [0,2*π) by finding where the sine function equals 1/2 (Quadrants I and II) and −1/2 (Quadrants III and IV).

x=π/6,(5*π)/6,(7*π)/6,(11*π)/6

6. Generalize the solution by adding multiples of the period π to the primary solutions, since the values are separated by exactly π

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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