Solve for x 3sec(x)^2-4=0

Problem

3*sec2(x)−4=0

Solution

1. Isolate the trigonometric term by adding 4 to both sides and then dividing by 3

3*sec2(x)=4

sec2(x)=4/3

2. Apply the reciprocal identity sec(x)=1/cos(x) to rewrite the equation in terms of cosine.

cos2(x)=3/4

3. Take the square root of both sides, remembering to include both the positive and negative roots.

cos(x)=±√(,3/4)

cos(x)=±√(,3)/2

4. Determine the reference angle by finding the value of x where cos(x)=√(,3)/2

x=π/6

5. Identify all solutions within the interval [0,2*π) where the cosine is √(,3)/2 or −√(,3)/2

x=π/6,(5*π)/6,(7*π)/6,(11*π)/6

6. Generalize the solution by adding multiples of the period π since the solutions are separated by π radians.

x=π/6+n*π

x=(5*π)/6+n*π

Final Answer

x=π/6+n*π,(5*π)/6+n*π

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