If 3 sin x + 3 sin 4x = sin y and 3 cos x + 3 cos 4x = cos y, then cos 3x?

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AAI ATC Junior Executive 21 Feb 2023 Shift 2 Official Paper
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  1. 17/18
  2. -17/18
  3. -1/18
  4. 1/18

Answer (Detailed Solution Below)

Option 2 : -17/18
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Detailed Solution

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Given:

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 .

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