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

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

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If sin(θ) = 0.6 and θ is acute, what is cos(θ)?

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What is the value of log₃(27)?

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What is the sum of all angles in a hexagon?

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What is the square root of 0.25?

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A sum of money triples itself in 12 years at simple interest. What is the rate of interest per annum?

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If two complementary angles differ by 30°, what are the angles?

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If x^3 - 3x^2 + 4 = 0, what is one root of the equation?

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The simple interest on Rs. 4000 at 5% per annum for 2 years is:

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

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