Question

\(sin(3sin^{-1}(\frac{1}{5})) =\)

• A. \(\frac{74}{125}\)
• B. \(\frac{71}{125}\)
• C. \(\frac{3}{5}\)
• D. \(\frac{1}{2}\)

Hint:

Apply the sine rule in two sub-triangles and use the ratio BD to DC.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Angles: \(B = 60^\circ\), \(C = 45^\circ\). D lies on BC with \(BD : DC = 1 : 3\). In triangle ABC, by the sine rule, \(\frac{AB}{AC} = \frac{\sin C}{\sin B} = \frac{\sin45^\circ}{\sin60^\circ} = \frac{\sqrt2}{\sqrt3}\).

Step 2: Sine rule in sub-triangles:
In triangle ABD: \(\frac{BD}{\sin\angle BAD} = \frac{AB}{\sin\angle ADB}\). In triangle ADC: \(\frac{DC}{\sin\angle CAD} = \frac{AC}{\sin\angle ADC}\). Since \(\sin\angle ADB = \sin\angle ADC\), dividing gives
\[ \frac{BD}{DC}\cdot\frac{\sin\angle CAD}{\sin\angle BAD} = \frac{AB}{AC} \]

Step 3: Solve:
\[ \frac{\sin\angle BAD}{\sin\angle CAD} = \frac{BD}{DC}\cdot\frac{AC}{AB} = \frac{1}{3}\cdot\frac{\sqrt3}{\sqrt2} = \frac{1}{\sqrt3\sqrt2} = \frac{1}{\sqrt6} \]

Final Answer:
The ratio is \(\frac{1}{\sqrt6}\), option (C). \[ \boxed{\frac{1}{\sqrt{6}}} \]