Topic Details (Notes format)

How to Use the Law of Cosines in Any Triangle

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.

Practice Questions

What is the value of log₃(27)?

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If the product of two numbers is 120 and their sum is 26, what are the numbers?

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The LCM of two numbers is 60, and their HCF is 5. If one of the numbers is 20, what is the other number?

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If a cylinder has a radius of 7 cm and height of 10 cm, what is its volume?

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If x = 3 and y = 4, what is the value of x^2 + y^2?

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What is the area of an equilateral triangle with side length 10 cm?

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If x^2 - 6x + 9 = 0, what is the value of x?

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If a + b = 10 and ab = 21, what is the value of a^2 + b^2?

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If 8x = 512, what is the value of x?

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If sin(A) = 3/5 and cos(B) = 5/13, where A and B are acute angles, what is sin(A+B)?

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