5Question: Compute \(\cos 45^\circ\) and use it to find the length of the diagonal of a square with side length \(7\) units.

["# How to Compute (\cos 45^\circ) and Find the Diagonal of a Square with Side 7 Units", "Understanding fundamental trigonometric values is essential in geometry, especially when calculating lengths such as diagonals in squares. One key value is (\cos 45^\circ), which plays a vital role in determining diagonal lengths efficiently. In this article, we’ll explore how to compute (\cos 45^\circ), apply it to calculate the diagonal of a square, and explain its practical application step by step.", "## What Is (\cos 45^\circ)?", "The cosine of (45^\circ) is a well-known value derived from the properties of an isosceles right triangle. In a right triangle where the two non-right angles are each (45^\circ), the sides are in the ratio (1 : 1 : \sqrt{2}). Specifically:", "[\n\cos 45^\circ = \frac{\ ext{adjacent}}{\ ext{hypotenuse}} = \frac{1}{\sqrt{2}} \approx 0.7071\n]", "This value is crucial because it allows us to relate the side length of a square to its diagonal using simple proportion.", "## Using (\cos 45^\circ) to Find the Diagonal of a Square", "Imagine a square with each side measuring (7) units. The diagonal of the square splits it into two identical 45-45-90 triangles. In such triangles:", "- The legs are the square’s sides (length (7)).\n- The diagonal is the hypotenuse.", "Using the cosine function, we can derive the diagonal length (d) as follows:", "[\n\cos 45^\circ = \frac{\ ext{side}}{\ ext{diagonal}} = \frac{7}{d}\n]", "Substitute (\cos 45^\circ = \frac{1}{\sqrt{2}}):", "[\n\frac{1}{\sqrt{2}} = \frac{7}{d}\n]", "Now solve for (d):", "[\nd = 7 \cdot \sqrt{2}\n]", "### Final Calculation:", "[\nd = 7\sqrt{2} \approx 9.90 \ ext{ units}\n]", "## Practical Summary", "- (\cos 45^\circ = \frac{1}{\sqrt{2}})\n- In a square of side (7):\n - Diagonal length = (7\sqrt{2})\n - Approximate value ≈ (9.90) units", "## Why This Matters", "Whether you're designing architecture, working in CAD software, or studying geometry, the relationship between angle measures and side-to-diagonal ratios simplifies complex calculations. Knowing (\cos 45^\circ) lets you quickly find square diagonals without messy Pythagorean theorem work (though (\ d = \sqrt{7^2 + 7^2} = \sqrt{98} = 7\sqrt{2}) confirms the same result).", "### Key Takeaway:", "- Use (\cos 45^\circ = \frac{1}{\sqrt{2}}) to efficiently compute the diagonal of a square.\n- For a square with side length (7), the diagonal is (7\sqrt{2}).\n- This technique saves time and reduces errors in geometric problem-solving.", "---", "Keywords: (\cos 45^\circ), cosine of 45 degrees, square diagonal formula, geometry tips, 7 unit square diagonal, trigonometry in geometry, diagonal of square calculator", "Meta Description: Learn how (\cos 45^\circ = \frac{1}{\sqrt{2}}) helps compute the diagonal of a square with side length 7 units using simple trigonometric principles. Step-by-step calculation included."]