Trigonometric Value Calculation

We need to find the value of the expression \(\sin \left( \frac{\theta}{2} \right) \sin \left( \frac{3\theta}{2} \right)\) given that \(\cos\theta = \frac{1}{3}\).

Applying Product-to-Sum Identity

We can use the product-to-sum trigonometric identity:

\( \sin(A) \sin(B) = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \)

Let \(A = \frac{3\theta}{2}\) and \(B = \frac{\theta}{2}\). Substituting these into the identity:

\( \sin \left( \frac{3\theta}{2} \right) \sin \left( \frac{\theta}{2} \right) = \frac{1}{2} \left[ \cos \left( \frac{3\theta}{2} - \frac{\theta}{2} \right) - \cos \left( \frac{3\theta}{2} + \frac{\theta}{2} \right) \right] \)

Simplify the terms inside the cosine functions:

\( = \frac{1}{2} [\cos(\theta) - \cos(2\theta)] \)

Calculating Cosine Double Angle

We are given \(\cos\theta = \frac{1}{3}\). We need to find \(\cos(2\theta)\).

Using the double angle identity \(\cos(2\theta) = 2\cos^2\theta - 1\):

\( \cos(2\theta) = 2 \left( \frac{1}{3} \right)^2 - 1 \)

\( \cos(2\theta) = 2 \left( \frac{1}{9} \right) - 1 \)

\( \cos(2\theta) = \frac{2}{9} - 1 \)

\( \cos(2\theta) = \frac{2 - 9}{9} = -\frac{7}{9} \)

Final Value Calculation

Now substitute the values of \(\cos\theta\) and \(\cos(2\theta)\) back into the expression from the product-to-sum step:

\( \sin \left( \frac{\theta}{2} \right) \sin \left( \frac{3\theta}{2} \right) = \frac{1}{2} [\cos(\theta) - \cos(2\theta)] \)

\( = \frac{1}{2} \left[ \frac{1}{3} - \left( -\frac{7}{9} \right) \right] \)

\( = \frac{1}{2} \left[ \frac{1}{3} + \frac{7}{9} \right] \)

Find a common denominator:

\( = \frac{1}{2} \left[ \frac{3}{9} + \frac{7}{9} \right] \)

\( = \frac{1}{2} \left[ \frac{10}{9} \right] \)

\( = \frac{5}{9} \)

The value of the expression is \(\frac{5}{9}\).