Question
Download Solution PDFसमीकरण \(\tan \ x = -\frac{1}{\sqrt 3} \) का प्रमुख हल ज्ञात कीजिए।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
समीकरण tan x = tan α के सामान्य हल को: x = nπ + α द्वारा ज्ञात किया जाता है जहाँ \(\alpha ∈ \left( { - \frac{π }{2},\frac{π }{2}} \right)\) और n ∈ Z है।
सूचना: एक त्रिकोणमितीय समीकरण के हल को प्रमुख हल कहा जाता है जिसके लिए 0 ≤ x < 2π है।
गणना:
दिया गया है: \(\tan \ x = -\frac{1}{\sqrt 3} \)
चूँकि हम जानते हैं कि, \(\tan \ \frac{5π}{6} = -\frac{1}{\sqrt 3} \)
⇒ \(\tan \ x = \tan \ \frac{5π}{6}\)
चूँकि हम जानते हैं कि, यदि tan x = tan α है, तो x = nπ + α है, जहाँ \(\alpha ∈ \left( { - \frac{π }{2},\frac{π }{2}} \right)\)और n ∈ Z है।
⇒ x = nπ + (5π/6) जहाँ n ∈ Z.
चूँकि हम जानते हैं कि, यदि एक त्रिकोणमितीय समीकरण के हल को प्रमुख हल कहा जाता है जिसके लिए 0 ≤ x < 2π है।
इसलिए, दिए गए समीकरण के प्रमुख हल x = 5π/6 और 11π/6 हैं।
अतः सही विकल्प 3 है।
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