If \(\sin(\theta) = \frac{3}{5}\), find \(\cos(\theta)\) for \(0 < \theta < \frac{\pi}{2}\).
["### Finding (\cos(\ heta)) When (\sin(\ heta) = \frac{3}{5}) for (0 < \ heta < \frac{\pi}{2})", "When solving trigonometric problems involving right triangles or angle relationships, understanding how to find one trigonometric function when another is known is essential. One common question is: If (\sin(\ heta) = \frac{3}{5}) and (0 < \ heta < \frac{\pi}{2}), what is (\cos(\ heta))?", "In the first quadrant, where (0 < \ heta < \frac{\pi}{2}), all trigonometric functions are positive. Given (\sin(\ heta) = \frac{\ ext{opposite}}{\ ext{hypotenuse}} = \frac{3}{5}), we can use the Pythagorean identity to find (\cos(\ heta) = \frac{\ ext{adjacent}}{\ ext{hypotenuse}}).", "#### Step-by-Step Solution Using the Pythagorean Identity", "1. Recall the identity:\n [\n \sin^2(\ heta) + \cos^2(\ heta) = 1\n ]", "2. Substitute the known value:\n [\n \left(\frac{3}{5}\right)^2 + \cos^2(\ heta) = 1\n ]\n [\n \frac{9}{25} + \cos^2(\ heta) = 1\n ]", "3. Solve for (\cos^2(\ heta)):\n [\n \cos^2(\ heta) = 1 - \frac{9}{25} = \frac{16}{25}\n ]", "4. Take the positive square root (since (\cos(\ heta) > 0) in Quadrant I):\n [\n \cos(\ heta) = \sqrt{\frac{16}{25}} = \frac{4}{5}\n ]", "#### Conclusion", "Therefore, when (\sin(\ heta) = \frac{3}{5}) and (0 < \ heta < \frac{\pi}{2}),\n[\n\cos(\ heta) = \frac{4}{5}.\n]", "This result is key in solving many trigonometric equations and applications, such as modeling waves, mechanics, or geometry problems involving right triangles.", "---", "Keywords: (\sin(\ heta) = \frac{3}{5}), (\cos(\ heta)), trigonometric identity, Pythagorean theorem, right triangle, first quadrant, inverse sine.", "Meta Description:\nLearn how to find (\cos(\ heta)) when (\sin(\ heta) = \frac{3}{5}) for (0 < \ heta < \frac{\pi}{2}) using the Pythagorean identity and quadrant analysis. Step-by-step solution with explanation."]
