Question
Download Solution PDFIf 3 sin x + 3 sin 4x = sin y and 3 cos x + 3 cos 4x = cos y, then cos 3x?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
3 sin x + 3 sin 4x = sin y and 3 cos x + 3 cos 4x = cos y
Concept:
Use trigonometric identity sin 2 x + cos 2 x = 1
And formula cos ( A - B ) = cos A cos B + sin A sin B
Calculation:
3 sin x + 3 sin 4x = sin y and 3 cos x + 3 cos 4x = cos y
After squaring both the sided add the given equations
( 3 sin x + 3 sin 4x )2 + ( 3 cos x + 3 cos 4x )2 = (sin y)2 + (cos y)2
⇒ ( 9 sin 2 x + 9 sin 2 4x + 18 sin x sin 4x ) + ( 9 cos 2 x + 9 cos 2 4x + 18 cos x cos 4x ) = 1
⇒ 9 ( sin 2 x + cos 2 x ) + 9 ( sin 2 4x + cos 2 4x ) + 18 ( 18 cos 4x cos x + sin 4x sin x ) = 1
⇒ 9 + 9 + 18 cos 3x = 1
⇒ 18 cos 3x = -17
⇒ cos 3x = \(\frac{-17}{18}\)
Hence the option (2) is correct .
Last updated on Jul 4, 2025
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