Question

If the range of $f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}, \, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite G.P., whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to ____.

Answer 1

The given function is:
$f(\theta) = \frac{\sin^4\theta + 3\cos^2\theta}{\sin^4\theta + \cos^2\theta}.$
Substitute $\cos^2\theta = x$, with $x \in [0, 1]$:
$f(x) = \frac{\sin^4\theta + 3x}{\sin^4\theta + x}.$
Simplify:
$f(x) = \frac{2x}{x + 1} + 1.$
The range of $f(x)$ can be computed as:
$f_{\min} = 1, \quad f_{\max} = 3.$
Thus:
$\alpha = 1, \quad \beta = 3.$
The infinite geometric series is:
$S = \frac{\text{first term}}{1 - \text{common ratio}}.$
Substitute:
$S = \frac{64}{1 - \frac{1}{3}} = \frac{64}{\frac{2}{3}} = 64 \cdot \frac{3}{2} = 96.$
Final Answer: 96.