Question
Download Solution PDFनिम्नलिखित संख्याओं पर विचार करें:
I. tan 22.5°
II. cot 22.5°
III. tan 22.5° - cot 22.5°
उपरोक्त में से कितनी अपरिमेय संख्याएँ हैं?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFव्याख्या:
(I) मान लीजिए tan 22.5° = tan 45/2 = t
⇒ tan45° = \(tan( 2 \times\frac{45}{2})\)
= \(\frac{2tan\frac{45}{2}}{1 -tan^2\frac{45}{2}}\)
⇒ 1 = \(\frac{2t}{1-t^2}\) ⇒ t2 + 2t -1 = 0
⇒ t = \(\frac{-2 ± √4+4}{2} = -1±√2\)
⇒ tan \(\frac{45}{2} = -1 ± √2 = -1 +√2\) (प्रथम चतुर्थांश में tan धनात्मक है)
यह एक अपरिमेय संख्या है।
(II) cot22.5° = \(\frac{1}{tan 22.5} = \frac{1}{-1 +√2} = 1+√2\)
यह भी एक अपरिमेय संख्या है
(III) tan22.5° - cot22.5° = -1+ √2 -(1 +√ 2)
= -2
यह एक अपरिमेय संख्या नहीं है
∴ विकल्प (ग) सही है।
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