Solve for x sec(x)-2=0

Problem

sec(x)−2=0

Solution

1. Isolate the trigonometric function by adding 2 to both sides of the equation.

sec(x)=2

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

1/cos(x)=2

3. Solve for cos(x) by taking the reciprocal of both sides.

cos(x)=1/2

4. Determine the reference angle by finding the value of x in the first quadrant where the cosine is 1/2

x=π/3

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/3 and x=2*π−π/3

x=π/3,(5*π)/3

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

x=π/3+2*π*n

x=(5*π)/3+2*π*n

Final Answer

sec(x)−2=0⇒x=π/3+2*π*n,(5*π)/3+2*π*n

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