Question

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

Answer 1

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

60 ≤ x ≤ 300

or

$60* \frac{ \pi }{180} = \frac{1}{3} \pi \\ \\ 300*\frac{ \pi }{180} = \frac{5}{3} \pi$

$\frac{1}{3} \pi \leq x \leq \frac{5}{3} \pi$