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We are given that \(\sin A = \frac{3}{4}\) and \(0 < A < \frac{\pi}{2}\), meaning \(A\) is in the first quadrant. We are tasked with finding \(\cos 2A\).

To solve this, we will use the double angle identity for cosine:

\[
\cos 2A = 1 - 2\sin^2 A
\]

First, we know \(\sin A = \frac{3}{4}\). Let's compute \(\sin^2 A\):

\[
\sin^2 A = \left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]

Now, substitute this into the identity:

\[
\cos 2A = 1 - 2\left(\frac{9}{16}\right) = 1 - \frac{18}{16} = 1 - \frac{9}{8} = \frac{-1}{8}
\]

Thus, the exact value of \(\cos 2A\) is:

\[
\cos 2A = \frac{-1}{8}
\]

Let me know if you would like further details or have any other questions.

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Here are 5 related questions for deeper understanding:
1. How do we find \(\cos A\) using \(\sin A\) if \(0 < A < \frac{\pi}{2}\)?
2. What are the other forms of the double angle identity for cosine?
3. Can we derive \(\cos 2A\) using the identity \(\cos 2A = 2\cos^2 A - 1\)?
4. How does the value of \(\cos 2A\) change if \(A\) were in a different quadrant?
5. How would you calculate \(\tan 2A\) from the given information?

**Tip:** Always check which quadrant the angle belongs to, as it affects the sign of trigonometric functions.

Given that sin A = 3/4 where 0 < A < π/2, find the exact value of cos 2A.

Solve for cos 2A given sin A = 3/4, where 0 < A < π/2. This problem uses the double angle identity for cosine, making it suitable for Grades 10-12 students studying trigonometry.

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