Question

Find the values of sin4 theta / sin theta (with proof)​

Answer 1

Step-by-step explanation:

To Find:-

• $\sf \: The \: value \: of \: \: \: \dfrac{sin4 \theta}{sin \theta}$

Formulas used:-

• sin(A + B) = sinA cosB + cosA sinB

• sin3θ= 3sinθ - 4sin³θ

• cos3θ = 4cos³θ - 3cosθ

• sin²θ = 1 - cos²θ

Solution:-

sin4θ = sin(3θ + θ)

= sin3θ cosθ + cos3θ sinθ

= (3sinθ - 4sin³θ) cosθ + (4cos³θ - 3cosθ) sinθ

= 3sinθcosθ - 4sin³θ cosθ + 4cos³θ sinθ - 3cosθ sinθ

= 3sinθcosθ - 3cosθ sinθ + 4cos³θ sinθ - 4sin³θ cosθ

= 4cos³θ sinθ - 4sin³θ cosθ

= sinθ(4cos³θ - 4sin²θ cosθ)

$\therefore \underline{ \: \: \underline{ \: \sf sin4 \theta = sin \theta(4 {cos}^{3} \theta - 4 {sin}^{2} \theta cos \theta) \: } \: \: }$

We need to find the value of $\sf \dfrac{sin4 \theta}{sin \theta}$

$\sf \dfrac{sin4 \theta}{sin \theta} = \dfrac{sin \theta(4 {cos}^{3} \theta - 4 {sin}^{2} \theta cos \theta)}{sin \theta}$

$\sf = 4 {cos}^{3} \theta - 4 {sin}^{2} \theta cos \theta$

$\sf = 4 {cos}^{3} \theta - 4(1 - {cos}^{2} \theta) cos \theta$

$\sf = 4 {cos}^{3} \theta - 4 cos \theta + 4cos^3 \theta$

$\sf = 8 {cos}^{3} \theta - 4 cos \theta$

$\underline{\boxed{\sf \therefore \dfrac{sin4 \theta}{sin \theta} = 8 {cos}^{3} \theta - 4 cos \theta}}$