Use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos C \), where \( a = 13 \), \( b = 15 \), \( C = 60^\circ \), \( \cos 60^\circ = 0.5 \).

["Using the Law of Cosines to Solve Triangles: A Complete Guide with ( a = 13 ), ( b = 15 ), and ( C = 60^\circ )", "When solving triangles in trigonometry, the Law of Cosines is an essential tool—especially when you know two sides and the included angle. In this article, we’ll explore how to apply the Law of Cosines using real values: ( a = 13 ), ( b = 15 ), and ( C = 60^\circ ), with ( \cos 60^\circ = 0.5 ). Whether you're working on geometry homework or engineering calculations, understanding this formula helps you find unknown sides and angles accurately.", "---", "### What is the Law of Cosines?", "The Law of Cosines generalizes the Pythagorean Theorem for any triangle, whether acute, right, or obtuse. It relates the lengths of the sides of a triangle to the cosine of one of its angles:", "[\nc^2 = a^2 + b^2 - 2ab\cos C\n]", "This equation allows you to find side ( c ) when you know sides ( a ) and ( b ), and the included angle ( C ). It’s especially useful when applying trigonometric principles beyond right-angled triangles.", "---", "### Step-by-Step Calculation: Finding Side ( c )", "Let’s apply the Law of Cosines using the given values:\n- ( a = 13 )\n- ( b = 15 )\n- ( C = 60^\circ )\n- ( \cos 60^\circ = 0.5 )", "Plug into the formula:", "[\nc^2 = 13^2 + 15^2 - 2(13)(15)(0.5)\n]", "Now compute each term:", "- ( 13^2 = 169 )\n- ( 15^2 = 225 )\n- ( 2 \ imes 13 \ imes 15 \ imes 0.5 = 2 \ imes 195 \ imes 0.5 = 195 )", "So,", "[\nc^2 = 169 + 225 - 195 = 394 - 195 = 199\n]", "Now take the square root to find ( c ):", "[\nc = \sqrt{199} \approx 14.106\n]", "> ✅ Final answer: ( c \approx 14.11 ) (rounded to two decimal places)", "---", "### Using the Law of Cosines to Find Angles", "You’re not limited to solving for side ( c ). You can also use the Law of Cosines to find angle ( A ) or ( B ) once one side is known. For example, once ( c = \sqrt{199} ), use the Law of Cosines again:", "[\na^2 = b^2 + c^2 - 2bc\cos A\n]", "Rearranged to solve for ( \cos A ):", "[\n\cos A = \frac{b^2 + c^2 - a^2}{2bc}\n]", "Substitute values:", "- ( b = 15 ), ( c^2 = 199 ), ( a = 13 )", "[\n\cos A = \frac{15^2 + 199 - 13^2}{2 \ imes 15 \ imes \sqrt{199}} = \frac{225 + 199 - 169}{30\sqrt{199}} = \frac{255}{30\sqrt{199}} = \frac{17}{2\sqrt{199}}\n]", "This gives the cosine of angle ( A ), which you can convert to degrees using inverse cosine:", "[\nA = \cos^{-1}\left( \frac{17}{2\sqrt{199}} \right)\n]", "Calculating numerically gives ( A \approx 49.1^\circ ), completing the triangle’s angle set.", "---", "### Applications of the Law of Cosines", "The Law of Cosines is widely used in:", "- Surveying and navigation to calculate distances across uneven terrains\n- Physics to resolve vector challenges where right angles don’t exist\n- Computer graphics and game design for accurate 3D environment modeling\n- Civil engineering for structural calculations involving non-right triangles", "---", "### Summary", "- The Law of Cosines: ( c^2 = a^2 + b^2 - 2ab\cos C )\n- Given: ( a = 13 ), ( b = 15 ), ( C = 60^\circ ), ( \cos 60^\circ = 0.5 )\n- Calculated: ( c \approx 14.11 )\n- Useful for solving unknown sides or angles in any triangle\n- Supports advanced applications in math, science, and engineering", "---", "### Key Takeaway", "Mastering the Law of Cosines opens the door to solving complex geometric problems efficiently—whether you’re calculating triangle dimensions in a classroom or applying real-world trigonometry in professional fields. Always start with known sides and included angles, then apply the formula step-by-step to uncover missing pieces of a triangle.", "---", "Ready to practice? Try solving another triangle using the Law of Cosines with different side lengths and angles to strengthen your skills!", "---", "Keywords: Law of Cosines, triangle calculation, trigonometry formula, solve triangles, use cosine law, ( c^2 = a^2 + b^2 - 2ab\cos C ), angle and side calculations, geometry solutions."]