To find the value of \sin \theta given \sec \theta = \frac{17}{12}, we can use trigonometric identities.

The identity for secant and cosine is:

\sec \theta = \frac{1}{\cos \theta}

Given that \sec \theta = \frac{17}{12}, we can find \cos \theta as follows:

\cos \theta = \frac{1}{\sec \theta} = \frac{1}{\frac{17}{12}} = \frac{12}{17}

Next, use the Pythagorean identity:

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

Substitute the value of \cos \theta:

\sin^2 \theta + \left(\frac{12}{17}\right)^2 = 1

Calculate the square of \cos \theta:

\left(\frac{12}{17}\right)^2 = \frac{144}{289}

Substitute back into the equation:

\sin^2 \theta = 1 - \frac{144}{289}

Simplify:

\sin^2 \theta = \frac{289}{289} - \frac{144}{289} = \frac{145}{289}

To find \sin \theta, take the square root:

\sin \theta = \sqrt{\frac{145}{289}} = \frac{\sqrt{145}}{17}

Therefore, the value of \sin \theta is \frac{\sqrt{145}}{17}, which corresponds to option (c).

Other options are incorrect because:

• \frac{13}{17} does not satisfy the Pythagorean identity when calculated with the given \sec \theta.
• \frac{12}{17} is the value of \cos \theta, not \sin \theta.
• \frac{\sqrt{170}}{17} is not derived from the given conditions.