Question

please solve it asap with proper explanation​

Answer 1

Step-by-step explanation:

Given :-

Sin³ θ Cos (3θ) + Cos³ θ Sin ( 3θ) = 3/8

To find :-

Find the value of Sin (4θ) ?

Solution :-

Given that :

Sin³ θ Cos (3θ) + Cos³ θ Sin ( 3θ) = 3/8

We know that

Cos 3θ = 4 Cos³ θ - 3 Cos θ

Sin 3θ = 3 Sin θ - 4 Sin³ θ

On Substituting these formulae in the given equation then

=>Sin³θ(4Cos³θ-3Cosθ)+Cos³θ(3Sinθ-4Sin³θ) = 3/8

=> 4Sin³θCos³θ- 3Sin³θCosθ +3Cos³θSinθ -4Sin³θCos³θ =3/8

=>(4Sin³θCos³θ-4Sin³θCos³θ) -3Sin³θCosθ +3Cos³θSinθ = 3/8

=> 0- 3Sin³θCosθ +3Cos³θSinθ = 3/8

=> - 3Sin³θCosθ +3Cos³θSinθ = 3/8

=> 3Cos³θSinθ - 3Sin³θCosθ = 3/8

=> 3( Cos³θSinθ - Sin³θCosθ) = 3/8

=> Cos³θSinθ - Sin³θCosθ = 3/(8×3)

=> Cos³θSinθ - Sin³θCosθ = 1/8

=> CosθSinθ ( Cos²θ- Sin²θ) = 1/8

We know that Cos²θ - Sin²θ = Cos 2θ

=> CosθSinθ( Cos 2θ) = 1/8

On multiplying with 2 both sides then

=> 2CosθSinθ( Cos 2θ) = 2×(1/8)

=> 2CosθSinθ( Cos 2θ) = 2/8

=> 2CosθSinθ( Cos 2θ) = 1/4

We know that

Sin 2A = 2 Sin A Cos A

=> (Sin2θ)( Cos 2θ) = 1/4

On multiplying with 2 both sides again then

=> 2[(Sin2θ)( Cos 2θ)] = 2×(1/4)

=> Sin 2(2θ) = 2/4

=> Sin 4θ = 1/2

Answer:-

The value of Sin 4θ for the given problem is 1/2

Used formulae:-

• Cos 3θ = 4 Cos³ θ - 3 Cos θ
• Sin 3θ = 3 Sin θ - 4 Sin³ θ

• Sin 2A = 2 Sin A Cos A

• Cos²θ - Sin²θ = Cos 2θ