Question

Find all solutions of the equation in the interval [0, 2π).

Answer 1

x = pi/6

x = 11pi/6

x = 5pi/6

x =7pi/6

Explanation:

2 sec^2 (x) + tan ^2 (x) -3 =0

We know tan^2(x) = sec^2 (x) -1

2 sec^2 (x) +sec^2(x) -1 -3 =0

Combine like terms

3 sec^2(x) -4 = 0

Add 4 to each side

3 sec^2 (x) = 4

Divide by 3

sec^2 (x) = 4/3

Take the square root of each side

sqrt(sec^2 (x)) = ±sqrt(4/3)

sec(x) = ±sqrt(4)/sqrt(3)

sec(x) = ±2 /sqrt(3)

Take the inverse sec on each side

sec^-1 sec(x) = sec^-1(±2 /sqrt(3))

x = pi/6 + 2 pi n where n is an integer

x = 11pi/6 + 2 pi n

x = 5pi/6 + 2 pi n

x =7pi/6 + 2 pi n

We only want the solutions between 0 and 2pi