A right triangle has legs of lengths \(6 \, \text{cm}\) and \(8 \, \text{cm}\). Compute the length of the hypotenuse and find the cosine of the angle opposite the shorter leg.

["A Right Triangle with Legs 6 cm and 8 cm: Compute the Hypotenuse and Cosine of the Opposite Angle", "In trigonometry, right triangles offer a foundational framework for understanding relationships between side lengths and angles. This article explores a right triangle with legs measuring 6 cm and 8 cm. We will compute the length of the hypotenuse using the Pythagorean theorem and determine the cosine of the angle opposite the shorter leg.", "---", "### Step 1: Identify the Triangle Dimensions", "Given a right triangle where:", "- Leg ( a = 6 , \ ext{cm} ) (shorter leg)\n- Leg ( b = 8 , \ ext{cm} ) (longer leg)\n- Hypotenuse ( c = ? )", "Since both legs are known, the hypotenuse can be found using the Pythagorean theorem:", "[\nc^2 = a^2 + b^2\n]", "Substitute the given values:", "[\nc^2 = 6^2 + 8^2 = 36 + 64 = 100\n]", "Take the positive square root:", "[\nc = \sqrt{100} = 10 , \ ext{cm}\n]", "Thus, the hypotenuse measures 10 cm, confirming this is a classic 6–8–10 Pythagorean triplet.", "---", "### Step 2: Determine the Angle Opposite the Shorter Leg", "Let angle ( \ heta ) be opposite the shorter leg (6 cm) and adjacent to the longer leg (8 cm). We aim to compute:", "[\n\cos(\ heta) = \frac{\ ext{adjacent}}{\ ext{hypotenuse}} = \frac{8}{10} = \frac{4}{5}\n]", "---", "### Step 3: Interpreting the Cosine Value", "The cosine of angle ( \ heta ) is:", "[\n\cos(\ heta) = \frac{4}{5} = 0.8\n]", "This means that in a right triangle with legs 6 cm and 8 cm, the cosine of the angle opposite the 6 cm leg is ( \frac{4}{5} ), demonstrating the direct application of right triangle functions.", "---", "### Conclusion", "For a right triangle with legs 6 cm and 8 cm:", "- The hypotenuse is ( 10 , \ ext{cm} ).\n- The cosine of the angle opposite the 6 cm leg is ( \frac{4}{5} ).", "Understanding these basic relationships is key to solving more complex problems in geometry and applied mathematics such as engineering, physics, and navigation.", "---", "Keywords:\nright triangle, legs 6 cm, legs 8 cm, hypotenuse length, cosine of angle, trigonometry, Pythagorean theorem, 6-8-10 triangle, cosine of angle opposite shorter leg, trigonometric ratios, geometry formulas."]