Question

If $\theta$ and $\varphi$ are the roots of the equation $8x^2+22x+5=0$ , then

• A. both $si{n}^{-1}\theta$ and $si{n}^{-1}\varphi$ are real
• B. both $se{c}^{-1}\theta$ and $se{c}^{-1}\varphi$ are real
• C. both $ta{n}^{-1}\theta$ and $ta{n}^{-1}\varphi$ are real
• D. None of these

Answer 1

Answer: (C)

$8x^3+22x+5=0\Rightarrow x=-\frac{1}{4},\frac{5}{2}$
$\because -1<-\frac{1}{4}<1$ and $-\frac{5}{2}<-1$
$\therefore si{n}^{-1}\left(-\frac{1}{4}\right)$ exists but $si{n}^{-1}\left(-\frac{5}{2}\right)$ does not exist
$se{c}^{-1}\left(-\frac{5}{2}\right)$ exists but $se{c}^{-1}\left(-\frac{1}{4}\right)$ does not exist,
$ta{n}^{-1}\left(-\frac{1}{4}\right)$ and $ta{n}^{-1}\left(-\frac{5}{2}\right)$ both exist