Solve for x (2cos(x)-1)(sin(x)-1)=0

Problem

(2*cos(x)−1)*(sin(x)−1)=0

Solution

1. Apply the Zero Product Property by setting each factor equal to zero.

(2*cos(x)−1)=0

(sin(x)−1)=0

2. Solve the first equation for cos(x) by adding 1 to both sides and dividing by 2

2*cos(x)=1

cos(x)=1/2

3. Find the general solution for x where cos(x)=1/2 using the unit circle.

x=π/3+2*k*π

x=(5*π)/3+2*k*π

4. Solve the second equation for sin(x) by adding 1 to both sides.

sin(x)=1

5. Find the general solution for x where sin(x)=1 using the unit circle.

x=π/2+2*k*π

6. Combine all solutions into a single set where k is any integer.

Final Answer

x=π/3+2*k*π,(5*π)/3+2*k*π,π/2+2*k*π

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