Question
Download Solution PDFअंतराल [0, π] में आने वाले समीकरण sin x + sin 5x = sin 3x के हलों की संख्या क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
- sin (A ± B) = sin A cos B ± sin B cos A.
- sin 2A + sin 2B = 2 sin (A + B) cos (A - B).
- cos (2nπ + θ) = cos θ.
गणना:
sin x + sin 5x = sin 3x
sin 2A + sin 2B = 2 sin (A + B) cos (A - B) का प्रयोग करने पर, हमें निम्न प्राप्त होता है:
2 sin 3x cos 2x = sin 3x
⇒ sin 3x (2 cos 2x - 1) = 0
⇒ sin 3x = 0 या 2 cos 2x - 1 = 0
स्थिति 1: sin 3x = 0 = sin nπ, n ∈ Z.
⇒
⇒
स्थिति 2: 2 cos 2x - 1 = 0
⇒
⇒
⇒
इसलिए, अंतराल [0, π] में x के 6 संभव मान हैं।
Additional Information
संबद्ध कोणों के लिए त्रिकोणमितीय अनुपात:
- sin (-θ) = -sin θ
- cos (-θ) = cos θ
- sin (2nπ + θ) = sin θ
- cos (2nπ + θ) = cos θ
- sin (nπ + θ) = (-1)n sin θ
- cos (nπ + θ) = (-1)n cos θ
= (-1)n cos θ = (-1)n (-sin θ)
Last updated on Jun 12, 2025
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