If $\theta$ and $\phi$ are the roots of the equation $8x^{2}+22x+5=0$, then
- both $\sin^{-1}{\theta}$ and $\sin^{-1}{\phi}$ are real
- both $\sec^{-1}{\theta}$ and $\sec^{-1}{\phi}$ are real
- both $\tan^{-1}{\theta}$ and $\tan^{-1}{\phi}$ are real
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None of these
Roots of 8x^2 + 22x + 5 = 0 are x = (-22 +/- sqrt(484 - 160)) / 16 = (-22 +/- sqrt(324)) / 16 = (-22 +/- 18) / 16. Roots are -4/16 = -0.25 and -40/16 = -2.5. For tan^-1(x) to be real, x can be any real number. For sin^-1(x) or sec^-1(x), the domain is restricted (-1 to 1 for sin, |x| >= 1 for sec). Since -2.5 is outside the domain for sin^-1 and sec^-1, only tan^-1 is valid.
First, we find the roots of the quadratic equation 8x^2 + 22x + 5 = 0 by factoring, which yields (4x + 1)(2x + 5) = 0. This gives the roots theta = -1/4 and phi = -5/2. For inverse trigonometric functions like arcsine to be real, the input values must fall within the interval from -1 to 1. While -1/4 is in this interval, -5/2 is not, so arcsine is not real for both roots. Because tangent and arctangent can take any real number as input, both arctan(-1/4) and arctan(-5/2) are valid and real. Therefore, both arctan(theta) and arctan(phi) are real.