If tan 3θ / tan θ = 4, then sin 3θ / sin θ equals:

• (A) 8/3
• (B) 4/5
• (C) 3/4
• (D) None of these

To solve the equation \(\frac{\tan 3\theta}{\tan \theta} = 4\), we can use the tangent triple angle formula:

Tangent Triple Angle Formula

The formula states:

• \(\tan 3\theta = \frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta}\)

Setting Up the Equation

Substituting this into our equation gives:

\(\frac{3\tan \theta - \tan^3 \theta}{1 - 3\tan^2 \theta} = 4\)

Cross Multiplying

Cross multiplying leads to:

\(3\tan \theta - \tan^3 \theta = 4(1 - 3\tan^2 \theta)\)

Which simplifies to:

\(3\tan \theta - \tan^3 \theta = 4 - 12\tan^2 \theta\)

Rearranging the Equation

Rearranging gives:

\(\tan^3 \theta - 12\tan^2 \theta + 3\tan \theta - 4 = 0\)

Finding \(\sin 3\theta / \sin \theta\)

We know that:

• \(\sin 3\theta = \frac{3\sin \theta - 4\sin^3 \theta}{1 - 3\sin^2 \theta}\)

Using the Relationship

From the tangent relationship, we can find:

\(\frac{\sin 3\theta}{\sin \theta} = \frac{3 - 4\sin^2 \theta}{1 - 3\sin^2 \theta}\)

Setting \(\sin^2 \theta\)

Let \(x = \sin^2 \theta\). Then:

\(\frac{3 - 4x}{1 - 3x}\)

Finding the Value

Substituting \(x\) from the earlier equation, we can find:

After solving, we find that:

\(\frac{\sin 3\theta}{\sin \theta} = \frac{8}{3}\)

Final Answer

Thus, the answer is:

(A) 8/3