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

cos(x)=−1/√(,2)

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

cos(x)=−√(,2)/2

4. Identify the reference angle by finding the value of x in the first quadrant where cos(x)=√(,2)/2

(x_ref)=π/4

5. Determine the quadrants where the cosine function is negative, which are Quadrant II and Quadrant III.

x=π−π/4=(3*π)/4

x=π+π/4=(5*π)/4

6. Generalize the solution by adding multiples of the period 2*π where n is any integer.

x=(3*π)/4+2*n*π

x=(5*π)/4+2*n*π

Final Answer

sec(x)=−√(,2)⇒x=(3*π)/4+2*n*π,(5*π)/4+2*n*π

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