Solve for x square root of 3sec(x)-2=0

Problem

√(,3)*sec(x)−2=0

Solution

1. Isolate the secant term by adding 2 to both sides of the equation.

√(,3)*sec(x)=2

2. Divide both sides by √(,3) to solve for sec(x)

sec(x)=2/√(,3)

3. Convert the secant function to its reciprocal, the cosine function, using the identity sec(x)=1/cos(x)

cos(x)=√(,3)/2

4. Determine the reference angle by finding the value of x in the first quadrant where the cosine is √(,3)/2

x=π/6

5. Identify all solutions within the interval [0,2*π) Since cosine is positive in the first and fourth quadrants, the solutions are x=π/6 and x=2*π−π/6=(11*π)/6

x=π/6,(11*π)/6

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

x=π/6+2*n*π

x=(11*π)/6+2*n*π

Final Answer

√(,3)*sec(x)−2=0⇒x=π/6+2*n*π,(11*π)/6+2*n*π

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