Solve for x sec(x) = square root of 2

Problem

sec(x)=√(,2)

Solution

1. Rewrite the equation in terms of the cosine function using the reciprocal identity sec(x)=1/cos(x)

1/cos(x)=√(,2)

2. Isolate the cosine function by taking the reciprocal of both sides.

cos(x)=1/√(,2)

3. Rationalize the denominator to identify the standard value on the unit circle.

cos(x)=√(,2)/2

4. Determine the reference angle in the first quadrant where the cosine is √(,2)/2

x=π/4

5. Identify all solutions within the standard interval [0,2*π) by considering the quadrants where cosine is positive (Quadrants I and IV).

x=π/4

x=2*π−π/4=(7*π)/4

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

x=π/4+2*n*π

x=(7*π)/4+2*n*π

Final Answer

sec(x)=√(,2)⇒x=π/4+2*n*π,(7*π)/4+2*n*π

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