Solve for x sec(x)=-(2 square root of 3)/3

Problem

sec(x)=−(2√(,3))/3

Solution

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

cos(x)=−3/(2√(,3))

2. Rationalize the denominator by multiplying the numerator and denominator by √(,3)

cos(x)=−(3√(,3))/(2*(3))

3. Simplify the fraction to find the standard value for the cosine function.

cos(x)=−√(,3)/2

4. Identify the reference angle by finding the value of θ such that cos(θ)=√(,3)/2

θ=π/6

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

x=π−π/6=(5*π)/6

x=π+π/6=(7*π)/6

6. Generalize the solution by adding multiples of the period 2*π to account for all possible values of x

x=(5*π)/6+2*π*n

x=(7*π)/6+2*π*n

Final Answer

sec(x)=−(2√(,3))/3⇒x=(5*π)/6+2*π*n,(7*π)/6+2*π*n

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