Solve for x sin(3x)=( square root of 2)/2

Problem

sin(3*x)=√(,2)/2

Solution

1. Identify the general solutions for the sine function where the value is √(,2)/2 On the unit circle, sin(θ)=√(,2)/2 at θ=π/4 and θ=(3*π)/4

2. Set up the equations for the argument 3*x by including the period 2*π*k where k is any integer.

3*x=π/4+2*π*k

3*x=(3*π)/4+2*π*k

3. Isolate x by dividing both sides of each equation by 3

x=π/12+(2*π*k)/3

x=(3*π)/12+(2*π*k)/3

4. Simplify the second expression by reducing the fraction (3*π)/12 to π/4

x=π/12+(2*π*k)/3

x=π/4+(2*π*k)/3

Final Answer

x=π/12+(2*π*k)/3,π/4+(2*π*k)/3

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