Question

(3) If 3 sino=4 cose, find 4 sin2O+2

Answer 1

Answer:

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Answer 2

Answer:

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Step-by-step explanation:

Given :

If

3 cos A + 4 sin A = 5,

Then, 4 cos A - 3 sin A = ?

Solution :

We know that,

( a + b )² = a² + 2ab +  b² ..(i)

( a - b )² = a² - 2ab + b² ..(ii)

sin² A + cos² A = 1

⇒ sin² A = 1 - cos² A ..(iii)

⇒ cos² A = 1 - sin² A ..(iv)

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⇒ 3 sin A + 4 cos A = 5

⇒ ( 3 sin A + 4 cos A )² = (5)²

⇒ ( 3 sin A )² + ( 4 cos A )² + 2 ( 3 sin A ) ( 4 cos A ) = 25 ( by (i) )

⇒ 9 sin² A + 16 cos² A + 24 sin A cos A = 25

⇒ 9 ( 1 - sin² A ) + 16( 1 - cos² A ) + 24 sin A cos A = 25  ( by (iii) & (iv) )

⇒ 9 - 9 sin² A + 16 - 16 cos² A + 24 sin A cos A = 25

⇒ 25 - 9 sin² A - 16 cos² A + 24 sin A cos A = 25

⇒ - 9 sin² A - 16 cos² A + 24 sin A cos A = 25 - 25 = 0

⇒ - ( 9 sin² A + 16 cos² A - 24 sin A cos A ) = 0

⇒ 9 sin² A + 16 cos² A - 24 sin A cos A = 0

⇒ ( 3 sin A )² + ( 4 cos A )² - 2 ( 3 sin A ) ( 4 cos A ) = 0

⇒ ( 4 cos A )² + ( 3 sin A )²  - 2 ( 3 sin A ) ( 4 cos A ) = 0

⇒ ( 4 cos A - 3 sin A)² = 0 ( by (ii) )

⇒ 4 cos A - 3 sin A = 0

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