Answer to Question #217198 in Trigonometry for Sanamtha

Question

Let x be an element of 0, [0, π]. We consider the inequalities {sinx>=[(2)^1/2]/2, 0<=cosx<(1/2). Which of the following statements is correct?

A. X is an element belonging to [π/4, π/3[;

B. X is an element belonging to ]π/4, 3π/4[

C. X is an element belonging to ]π/3, π]

D. X is an element belonging to ]π/3, π/2]

Answer 1

Let us consider the system of inequalities $\begin{cases} \sin x\ge\frac{\sqrt{2}}{2},\\ 0\le \cos x<\frac{1}{2}\end{cases}.$

The solutions of the inequality $\sin x\ge\frac{\sqrt{2}}{2}$ on the interval $[0,\pi]$ are all $x\in [\frac{\pi}{4},\frac{3\pi}{4}].$ The solutions of the inequality $0\le \cos x<\frac{1}{2}$ on the interval $[0,\pi]$ are all $x\in ]\frac{\pi}{3},\frac{\pi}{2}].$ Taking into account that $[\frac{\pi}{4},\frac{3\pi}{4}]\cap]\frac{\pi}{3},\frac{\pi}{2}]=]\frac{\pi}{3},\frac{\pi}{2}],$ we conclude that the solutions of the system are all elements $x$ belonging to $]\frac{\pi}{3},\frac{\pi}{2}].$

Answer: D