Problem 37

First, we use the cofunction identity to rewrite $\sin 2x$ as $\cos (\frac{\pi}{2} - 2x)$. Then, we use the double angle formula to express $\cos (\frac{\pi}{2} - 2x)$ as $1 - 2\sin^2(\frac{\pi}{4} - x)$.

Now, we can substitute the given value $\sin (\frac{\pi}{4} - x) = \frac{3}{5}$ into the above expression:

$$\sin 2x = \cos (\frac{\pi}{2} - 2x) = 1 - 2\sin^2(\frac{\pi}{4} - x) = 1 - 2(\frac{3}{5})^2 = \boxed{\frac{7}{25}}$$

The solution is presented step-by-step by first applying the cofunction identity, followed by the double angle formula, and finally substituting the given value to find the answer.

We know that $\sin(\frac{\pi}{4} - x) = \frac{3}{5}$. To find the value of $\sin 2x$, we can utilize the double angle identity for sine. However, it would be more convenient to first convert $\sin 2x$ into a cosine function using the cofunction identity.

The cofunction identity states that $\sin(\frac{\pi}{2} - \theta) = \cos \theta$. Applying this to our problem, we get:

$$\sin 2x = \cos(\frac{\pi}{2} - 2x).$$

Now, we can use the double angle identity for cosine, which states that $\cos 2\theta = 1 - 2\sin^2 \theta$. Setting $\theta = \frac{\pi}{4} - x$, we have:

$$\cos(2(\frac{\pi}{4} - x)) = 1 - 2\sin^2(\frac{\pi}{4} - x).$$

Since $\sin(\frac{\pi}{4} - x) = \frac{3}{5}$, we can substitute this value in:

$$\cos(2(\frac{\pi}{4} - x)) = 1 - 2\left(\frac{3}{5}\right)^2 = 1 - \frac{18}{25} = \frac{7}{25}.$$

Now, recall that $\sin 2x = \cos(\frac{\pi}{2} - 2x)$. To make the expressions match, we can note that $\frac{\pi}{2} - 2x = 2(\frac{\pi}{4} - x)$, so:

$$\sin 2x = \cos(2(\frac{\pi}{4} - x)) = \frac{7}{25}.$$

Therefore, the value of $\sin 2x$ is $\boxed{\frac{7}{25}}$.