Solve for x 4sin(x)^2-4sin(x)+1=0

Problem

4*sin2(x)−4*sin(x)+1=0

Solution

1. Identify the equation as a quadratic in terms of sin(x)

2. Substitute u=sin(x) to rewrite the equation as a standard quadratic.

4*u2−4*u+1=0

3. Factor the quadratic expression, noting that it is a perfect square trinomial.

(2*u−1)2=0

4. Solve for u by taking the square root of both sides.

2*u−1=0

u=1/2

5. Back-substitute sin(x) for u to find the values of x

sin(x)=1/2

6. Determine the general solution for x based on the unit circle, where the sine function equals 1/2 at π/6 and (5*π)/6

x=π/6+2*n*π

x=(5*π)/6+2*n*π

Final Answer

x=π/6+2*n*π,(5*π)/6+2*n*π

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