Subject: Mathematics
Book: Maths Mastery
The Law of Cosines extends the Pythagorean theorem to non-right triangles: c² = a² + b² – 2ab cos(C), where C is the angle opposite side c. If you know two sides and the included angle, you can find the third side; or if you know three sides, you can find an angle. For instance, if a=7, b=5, and angle C=60°, then c² = 7² + 5² – 2×7×5×cos(60°)= 49 + 25 – 70×0.5=49 + 25 – 35=39, so c=√39. This formula solves oblique triangles, essential in astronomy, land surveying, or advanced geometry proofs. Grasping the Law of Cosines complements the Law of Sines to solve any general triangle scenario.
If a:b = 5:7 and b:c = 6:11, what is a:c?
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