In a triangle, the lengths of two sides are 13 cm and 15 cm. If the angle between them is 60°, what is the length of the third side?
["Title: Use the Law of Cosines to Find the Third Side of a Triangle with Given Sides and Angle", "Meta Description:\nDiscover how to calculate the third side of a triangle when two sides and the included angle are known. Learn the step-by-step solution using the Law of Cosines with sides 13 cm and 15 cm and an angle of 60°.", "---", "### Solving for the Third Side in a Triangle with a 60° Angle", "When you know the lengths of two sides of a triangle and the measure of the angle between them, finding the third side is straightforward using the Law of Cosines. In this article, we’ll explain how to calculate the missing side when two sides and the included angle are given.", "#### Given:\n- Side a = 13 cm\n- Side b = 15 cm\n- Included angle C = 60°", "#### Goal:\nFind the length of side c, opposite angle C.", "---", "### What is the Law of Cosines?", "The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula is:", "[\nc^2 = a^2 + b^2 - 2ab\cos(C)\n]", "This formula works for any triangle, but it’s especially useful when the angle between sides a and b is known — that’s your included angle.", "---", "### Step-by-Step Calculation", "1. Plug known values into the formula:\n[\nc^2 = 13^2 + 15^2 - 2(13)(15)\cos(60^\circ)\n]", "2. Calculate squares:\n[\n13^2 = 169, \quad 15^2 = 225\n]\nSo,\n[\nc^2 = 169 + 225 - 2(13)(15)\cos(60^\circ)\n]\n[\nc^2 = 394 - 390\cos(60^\circ)\n]", "3. Use the known cosine value:\n[\n\cos(60^\circ) = 0.5\n]\nSubstitute:\n[\nc^2 = 394 - 390(0.5) = 394 - 195 = 199\n]", "4. Take the square root:\n[\nc = \sqrt{199} \approx 14.11 \ ext{ cm}\n]", "---", "### Final Answer:", "The length of the third side is approximately 14.11 cm. For maximum precision, use the exact form ( \sqrt{199} ) cm when exact values matter.", "---", "### Why This Matters", "Understanding how to apply the Law of Cosines helps in fields like architecture, engineering, and geography, where precise distance calculations are essential. Whether you're designing a roof, mapping terrain, or modeling structures, knowing how to compute a triangle’s missing side empowers accurate problem-solving.", "---", "#### Quick Recap:\n- Given: Two sides (13 cm, 15 cm) and angle (60°) between them\n- Use Law of Cosines: ( c^2 = a^2 + b^2 - 2ab\cos(C) )\n- Calculate step-by-step → Get ( c = \sqrt{199} \approx 14.11 \ ext{ cm} )", "---", "Remember: When two sides and the included angle are known, the Law of Cosines is your most reliable tool for finding the third side.", "---", "Keywords: Law of Cosines, triangle side length formula, find third side given two sides and included angle, 13 cm and 15 cm triangle with 60° angle, law of cosines example, triangle calculations, geometry problems.", "---", "By mastering this technique, you’ll efficiently solve countless triangle problems — whether in math class or real-world applications."]
