Sec inverse x greater than or equal to pi/4 than x?

To solve the inequality \(\sec^{-1}(x) \geq \frac{\pi}{4}\), we first need to understand what \(\sec^{-1}(x)\) represents. The secant inverse function, \(\sec^{-1}(x)\), gives us the angle whose secant is \(x\).

Understanding the Secant Function

The secant function is defined as:

• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

This means that \(\sec^{-1}(x)\) is defined for \(x \leq -1\) or \(x \geq 1\). The range of \(\sec^{-1}(x)\) is from \(0\) to \(\frac{\pi}{2}\) and from \(\frac{\pi}{2}\) to \(\pi\).

Finding the Values of x

Next, we need to determine when \(\sec^{-1}(x) \geq \frac{\pi}{4}\). The angle \(\frac{\pi}{4}\) corresponds to a secant value of:

• \(\sec\left(\frac{\pi}{4}\right) = \sqrt{2}\)

Thus, the inequality \(\sec^{-1}(x) \geq \frac{\pi}{4}\) translates to:

• \(x \leq -\sqrt{2}\) or \(x \geq \sqrt{2}\)

Final Answer

In conclusion, the solution to the inequality \(\sec^{-1}(x) \geq \frac{\pi}{4}\) is:

• x ≤ -√2
• x ≥ √2