Solve for x sec(4x)=2

Problem

sec(4*x)=2

Solution

1. Rewrite the equation using the reciprocal identity sec(θ)=1/cos(θ)

1/cos(4*x)=2

2. Invert both sides of the equation to solve for the cosine term.

cos(4*x)=1/2

3. Determine the general solutions for the angle 4*x by identifying where the cosine function equals 1/2 on the unit circle.

4*x=π/3+2*n*π

4*x=−π/3+2*n*π

4. Isolate x by dividing all terms in both general solution equations by 4

x=π/12+(n*π)/2

x=−π/12+(n*π)/2

Final Answer

sec(4*x)=2⇒x=±π/12+(n*π)/2,n∈ℤ

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