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

Problem

4*sin2(x)−3=0

Solution

1. Isolate the trigonometric term by adding 3 to both sides and then dividing by 4

4*sin2(x)=3

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)

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 the interval [0,2*π) by finding where the sine function equals √(,3)/2 or −√(,3)/2

x=π/3,(2*π)/3,(4*π)/3,(5*π)/3

5. Generalize the solution by adding multiples of 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

x=π/3+n*π,(2*π)/3+n*π

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