Assume that theta is a positive acute angle. Given: cos theta = 15/17. Find: sin 2t...

1. Identify the goal: The problem asks us to find the value of $\sin 2\theta$ given that $\cos \theta = \frac{15}{17}$ and $\theta$ is a positive acute angle.

1. Recall the double angle identity for sine: The double angle identity for sine is $\sin 2\theta = 2 \sin \theta \cos \theta$.

3. Determine the value of $\sin \theta$: We are given $\cos \theta = \frac{15}{17}$. We can use the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ to find $\sin \theta$.
$$\sin^2 \theta + \left(\frac{15}{17}\right)^2 = 1$$
$$\sin^2 \theta + \frac{225}{289} = 1$$
$$\sin^2 \theta = 1 - \frac{225}{289}$$
$$\sin^2 \theta = \frac{289 - 225}{289}$$
$$\sin^2 \theta = \frac{64}{289}$$
Since $\theta$ is a positive acute angle, $\sin \theta$ must be positive. Therefore, we take the positive square root:
$$\sin \theta = \sqrt{\frac{64}{289}} = \frac{8}{17}$$

4. Substitute the values of $\sin \theta$ and $\cos \theta$ into the double angle identity: Now we have $\sin \theta = \frac{8}{17}$ and $\cos \theta = \frac{15}{17}$. We can substitute these values into the formula $\sin 2\theta = 2 \sin \theta \cos \theta$.
$$\sin 2\theta = 2 \left(\frac{8}{17}\right) \left(\frac{15}{17}\right)$$

5. Calculate the final answer:
$$\sin 2\theta = \frac{2 \times 8 \times 15}{17 \times 17}$$
$$\sin 2\theta = \frac{240}{289}$$