Question

If θ is an acute angle and 2sin θ=1, then find the value of θ and hence find the value of 4sin³θ - 3sin θ​

Answer 1

Answer:

θ = 30°

4Sin³ θ - 3Sin θ = -1

Step-by-step explanation:

We are given,

θ is an acute angle and 2Sin θ = 1

Now,

2Sin θ = 1

Sin θ = (1/2)

But we know that,

Sin 30° = (1/2)

Thus,

Sin θ = Sin 30°

θ = 30°

Now,

We can find 4Sin³ θ - 3Sin θ, by 2 ways, either directly Substituting it or using the identity.

Substitution method

4Sin³ 30° - 3Sin 30°

= 4(Sin 30°)³ - 3(Sin 30°)

= 4(1/2)³ - 3(1/2)

= 4(1/8) - 3(1/2)

= (1/2) - (3/2)

= (1 - 3)/2

= (-2/2)

= -1

Identity method

We know that,

4Sin³ θ - 3Sin θ = -(3Sin θ - 4Sin³ θ)

And we know that Identity as

3Sin θ - 4Sin³ θ = Sin 3θ

Thus,

4Sin³ θ - 3Sin θ = -(3Sin θ - 4Sin³ θ)

Using identity,

4Sin³ θ - 3Sin θ = -(Sin 3θ)

4Sin³ θ - 3Sin θ = -(Sin 3(30°))

4Sin³ θ - 3Sin θ = -(Sin 90°)

4Sin³ θ - 3Sin θ = -(1)

4Sin³ θ - 3Sin θ = -1

Hence,

1) θ = 30°

2) 4Sin³ θ - 3Sin θ = -1

Hope it helped and believing you understood it........All the best

Answer 2

2sin∅ = 1 => sin∅ =1/2 => sin∅ = sin 30 => ∅ = 30

(u can also use inverse function here )

•°• 4 sin³30 - 3 sin 30 =( 4* 1/2*1/2*1/2 )- 3/2

= 1/2 - 3/2=-2/2= -1