Hisoblang: \( 3 \operatorname{arctg} 5+3 \operatorname{arctg} \frac{1}{5} \) \( \begin{array}{llll}\text { A) } \frac{\pi}{6} & \text { B) } \frac{\pi}{2} & \text { C) } \frac{3 \pi}{2} & \text { D) } \frac{5 \pi}{3}\end{array} \)

Bu ifodani hisoblash uchun, ilgari ko'p ma'lumot bo'lgan arktangens qoidasidan foydalanishimiz mumkin. Ma'lumki, \( \tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)} \) formula orqali arktangenslar qo'shilishi natijalaridan foydalanamiz. Shu bilan birga, \( \tan(\operatorname{arctg} x) = x \) ekanligini yodda tuting. Bunda: \[ a = 3 \operatorname{arctg} 5 \quad \text{va} \quad b = 3 \operatorname{arctg} \frac{1}{5} \] Ularning tangensini hisoblaymiz: \[ \tan(a) = 5 \quad \text{va} \quad \tan(b) = \frac{1}{5} \] Endi formulani qo'llaymiz: \[ \tan(a + b) = \frac{5 + \frac{1}{5}}{1 - 5 \cdot \frac{1}{5}} = \frac{5 + 0.2}{1 - 1} = \text{undefined} \] Bu natija \( \frac{\pi}{2} \) ga olib boradi, chunki \( \tan\) qiymati \( +\infty \) ga o'tadi. Shuning uchun, to'g'ri javob: \[ \text{B) } \frac{\pi}{2} \]