Solve for x sec(x)=2

Problem

sec(x)=2

Solution

1. Rewrite the equation using the reciprocal identity for the secant function, where sec(x)=1/cos(x)

1/cos(x)=2

2. Solve for cos(x) by taking the reciprocal of both sides of the equation.

cos(x)=1/2

3. Determine the reference angle by finding the value in the first quadrant where the cosine is 1/2

x=π/3

4. Identify all solutions within the standard interval [0,2*π) Since cosine is positive in the first and fourth quadrants, the solutions are x=π/3 and x=2*π−π/3=(5*π)/3

x=π/3,(5*π)/3

5. Generalize the solution to include all possible rotations by adding multiples of the period 2*π where n is any integer.

x=π/3+2*n*π

x=(5*π)/3+2*n*π

Final Answer

sec(x)=2⇒x=π/3+2*n*π,(5*π)/3+2*n*π

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