Solve for x sec(x) = square root of 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. Isolate the cosine function by taking the reciprocal of both sides of the equation.

cos(x)=1/√(,2)

3. Rationalize the denominator to express the value in a standard form found on the unit circle.

cos(x)=√(,2)/2

4. Identify the reference angle in the first quadrant where the cosine is √(,2)/2 which is x=π/4

x=π/4

5. Determine the general solution by considering the quadrants where cosine is positive (Quadrants I and IV) and the periodicity of the cosine function, which is 2*π*n for any integer n

x=π/4+2*π*n

x=−π/4+2*π*n

Final Answer

sec(x)=√(,2)⇒x=±π/4+2*π*n

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