Given that \(\cos \frac{a}{2} = \frac{\sqrt{51}}{12}\), we need to find \(\sin \frac{a}{2}\).

We can use the Pythagorean identity for trigonometric functions:

\[
\sin^2 \theta + \cos^2 \theta = 1
\]

Substituting \(\theta = \frac{a}{2}\):

\[
\sin^2 \frac{a}{2} + \cos^2 \frac{a}{2} = 1
\]

We already have \(\cos \frac{a}{2} = \frac{\sqrt{51}}{12}\). Therefore,

\[
\sin^2 \frac{a}{2} + \left(\frac{\sqrt{51}}{12}\right)^2 = 1
\]

Simplifying:

\[
\sin^2 \frac{a}{2} + \frac{51}{144} = 1
\]

Subtract \(\frac{51}{144}\) from both sides:

\[
\sin^2 \frac{a}{2} = 1 - \frac{51}{144}
\]

Convert 1 to \(\frac{144}{144}\):

\[
\sin^2 \frac{a}{2} = \frac{144}{144} - \frac{51}{144} = \frac{93}{144}
\]

Simplify the fraction:

\[
\sin^2 \frac{a}{2} = \frac{31}{48}
\]

Taking the square root of both sides:

\[
\sin \frac{a}{2} = \sqrt{\frac{31}{48}}
\]

Simplify the square root:

\[
\sin \frac{a}{2} = \frac{\sqrt{31}}{\sqrt{48}} = \frac{\sqrt{31}}{4\sqrt{3}} = \frac{\sqrt{31}}{4\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{93}}{12}
\]

So, \(\sin \frac{a}{2} = \frac{\sqrt{93}}{12}\).

Would you like more details on any step, or do you have any other questions?

Here are 5 related questions:

1. How do you find \(\tan \frac{a}{2}\) given \(\cos \frac{a}{2}\)?
2. What is the relationship between \(\sin \frac{a}{2}\) and \(\cos \frac{a}{2}\) using half-angle identities?
3. How can you verify \(\sin \frac{a}{2} + \cos \frac{a}{2} = 1\) for specific angles?
4. How do you solve trigonometric equations involving half-angles?
5. What are the applications of half-angle identities in integration?

**Tip:** Always simplify square roots in trigonometric identities to make calculations easier.

Learn how to find sin(a/2) when given cos(a/2) = sqrt(51)/12 using the Pythagorean identity and trigonometric calculations. Detailed step-by-step solution provided.

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