Thus, \(\cos 60^\circ\) is \(\boxed{\frac{1}{2}}\).Question: Find $\tan \theta$ where $\theta$ is the angle between the line of sight from a weather balloon at $(2, 3)$ to a cloud at $(5, 11)$ and the horizontal axis.

["Understanding the Angle of Elevation: How to Find (\ an \ heta) for a Weather Balloon", "When tracking a weather balloon rising from a known point to observe clouds, understanding the angle it forms with the horizontal is key for accurate meteorological readings. Take the example of a balloon located at coordinate ((2, 3)) lifting toward a cloud at ((5, 11)). The angle (\ heta) between the balloon’s line of sight and the horizontal can reveal both direction and elevation behavior—insights essential for weather monitoring and atmospheric studies.", "In this article, we explore how to calculate (\ an \ heta), the tangent of the angle (\ heta), using basic coordinate geometry. This simple trigonometric ratio connects the horizontal and vertical changes, offering a clear, measurable way to evaluate the balloon’s incline.", "---", "### Step 1: Visualize the Coordinates", "We begin by placing the balloon’s starting point at ((2, 3)) and the cloud at ((5, 11)). To analyze the slope of the line connecting these points—or equivalently, the angle of elevation from horizontal—we compute the horizontal and vertical displacements:", "- Horizontal change (Δx):\n [\n \Delta x = 5 - 2 = 3\n ]", "- Vertical change (Δy):\n [\n \Delta y = 11 - 3 = 8\n ]", "These differences represent the rise over run from the balloon’s position to the cloud’s position.", "---", "### Step 2: Relate Changes to the Tangent Function", "By definition, the tangent of the angle (\ heta) between the line of sight and the horizontal axis is given by:", "[\n\ an \ heta = \frac{\ ext{opposite}}{\ ext{adjacent}} = \frac{\Delta y}{\Delta x}\n]", "Substitute the values calculated:", "[\n\ an \ heta = \frac{8}{3}\n]", "This ratio captures the slope of the line connecting the two points and directly defines (\ an \ heta).", "---", "### Step 3: Final Result", "Thus, the tangent of the angle (\ heta) is:", "[\n\ an \ heta = \frac{8}{3}\n]", "This precise value quantifies the steepness of the balloon’s line of sight and supports accurate interpretation of vertical motion in meteorology.", "---", "Summary:\nFor a weather balloon at ((2, 3)) observing a cloud at ((5, 11)), the angle (\ heta) formed with the horizontal axis satisfies:", "[\n\ an \ heta = \frac{\Delta y}{\Delta x} = \frac{8}{3}\n]", "Understanding this trigonometric relationship simplifies elevation angle analysis, enhancing data accuracy in atmospheric research.", "If you’re measuring angles for balloon tracking or similar applications, remember:\n[\n\ an \ heta = \frac{\ ext{rise}}{\ ext{run}} = \frac{y_2 - y_1}{x_2 - x_1}\n]\nThis formula is foundational in geometry, physics, and meteorology.", "---", "Conclusion:\nFind (\ an \ heta) by computing vertical vs. horizontal change—here,\n[\n\ an \ heta = \boxed{\frac{8}{3}}\n]\nenabling precise angular analysis in weather balloon observations and beyond."]