Because $ \tan \theta = \frac{5}{12} \Rightarrow $ opposite = 5, adjacent = 12, hyp = 13. Then $ \sin \theta = \frac{5}{13} $, $ \cos \theta = \frac{12}{13} $, so $ \sin 2\theta = 2 \sin\theta \cos\theta = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169} $.

["Understanding Trigonometric Identities: How $ \ an \ heta = \frac{5}{12} $ Leads to Key Values and $ \sin 2\ heta = \frac{120}{169} $", "When learning trigonometry, one of the foundational principles is how the tangent ratio connects directly to the opposite, adjacent, and hypotenuse sides of a right triangle. Given the equation", "$$\n\ an \ heta = \frac{5}{12},\n$$", "we interpret this as the ratio of the opposite side to the adjacent side:\nopposite = 5, adjacent = 12. Using the Pythagorean theorem, we compute the hypotenuse:\n$$\n\ ext{hyp} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13.\n$$", "With this triangle established—opposite = 5, adjacent = 12, hypotenuse = 13—we can now determine the sine and cosine of angle $ \ heta $:", "$$\n\sin \ heta = \frac{\ ext{opposite}}{\ ext{hyp}} = \frac{5}{13},\n$$\n$$\n\cos \ heta = \frac{\ ext{adjacent}}{\ ext{hyp}} = \frac{12}{13}.\n$$", "From these fundamental values, we compute the double-angle identity for sine:", "$$\n\sin 2\ heta = 2 \sin \ heta \cos \ heta.\n$$", "Substituting the known values:", "$$\n\sin 2\ heta = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{2 \cdot 5 \cdot 12}{13 \cdot 13} = \frac{120}{169}.\n$$", "This elegant derivation showcases how basic trigonometric ratios lead naturally to deeper identities and exact computations. The result $ \sin 2\ heta = \frac{120}{169} $ is both precise and widely applicable in fields such as physics, engineering, and signal processing, where double-angle formulas are essential.", "In summary, recognizing $ \ an \ heta $ allows us to unlock the full suite of trigonometric values through the Pythagorean theorem and essential definitions. This step-by-step approach reinforces the central role of triangles and identities in mastering trigonometry. Whether solving angles, modeling waves, or analyzing forces, these relationships are indispensable tools in mathematical reasoning."]