Question

Solve the following inequality. Express the exact answer in interval notation, restricting your attention to 0 ≤ x ≤ 2π.

sec^2(x) ≤ 4

Answer 1

`sec^2(x) \leq 4 \\\\ (1)/(cos^2(x)) \leq 4 \\\\ (1)/(cos^2(x))-4\leq 0 \\\\ (1-4cos^2(x))/(cos^2(x)) \leq 0 \\\\ 1-4cos^2(x)\leq 0*cos^2(x) \\\\1-4cos^2(x) \leq 0 \\\\-4cos^2(x) \leq -1 \\\\cos^2(x) \leq (-1)/(-4) \\ \\ cos^2(x)\geq (1)/(4) \\ \\ √(cos^2(x)) \geq \sqrt{ (1)/(4)} \\\\cos(x) \geq (1)/(2) \\\\ x\geq60 \\------------- \\\\ \theta_I=60 \\\\ \theta_(IV)=360-60 \\ \theta_(IV)=300`

60 ≤ x ≤ 300

or


`60* ( \pi )/(180) = (1)/(3) \pi \\ \\ 300*( \pi )/(180) = (5)/(3) \pi`


`(1)/(3) \pi \leq x \leq (5)/(3) \pi`