Question

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

• A. both ${\sin }^{-1}\theta$ and ${\sin }^{-1}\varphi$ are real
• B. both ${\sec }^{-1}\theta$ and ${\sec }^{-1}\varphi$ are real
• C. both ${\tan }^{-1}\theta$ and ${\tan }^{-1}\varphi$ are real
• D. None of these

Answer 1

The correct option is C both ${\tan }^{-1}\theta$ and ${\tan }^{-1}\varphi$ are real

Solving given quadratic eq., we get

$8x^2+22x+5=0$

$⟹(4x+1)(2x+5)=0$

$⟹x=\frac{-1}{4},\frac{-5}{2}$

Hence these values are not in the range of $se{c}^{-1}x$ and $si{n}^{-1}x$

So, $ta{n}^{-1}\theta$ and $ta{n}^{-1}\varphi$ are real