Functions
Greatest Integer / Summation
Grade Class 12
Question:
Let $f(\theta) = \dfrac{1}{1+\cot^x\theta}$ (x-th power). Find $S = \displaystyle\sum_{\theta=1^\circ}^{89^\circ} f(\theta)$. Find $[S]$ (greatest integer).
Step-by-Step Solution
Key Concept: King's property: $f(\theta)+f(90°-\theta)=1$; pair up terms for $\theta=1°$ to $89°$ (44 pairs + middle $\theta=45°$).
$f(90°-\theta)=\frac{1}{1+\tan^x\theta}=1-f(\theta)$. Pairing: $S=\sum_{\theta=1}^{89}f(\theta)=44\cdot 1+f(45°)=44+\frac{1}{2}=44.5$. $[S]=44$.
Correct Answer: 44