Law of Cosines

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What is the Law of Cosines?

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The Law of Cosines states that in any triangle, the square of the length of one side (c) is equal to the sum of the squares of the lengths of the other two sides (a and b) minus twice the product of those two sides multiplied by the cosine of the included angle (C). Formula: c² = a² + b² - 2ab cos(C).

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When should you use the Law of Cosines instead of the Law of Sines?

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Use the Law of Cosines when you have two sides and the included angle (SAS) or all three sides (SSS) of a triangle. The Law of Sines is used when you have two angles and one side (ASA or AAS) or two sides and a non-included angle (SSA).

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What is the formula for the Law of Cosines?

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The Law of Cosines can be expressed as: c² = a² + b² - 2ab cos(C), a² = b² + c² - 2bc cos(A), b² = a² + c² - 2ac cos(B).

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What does the term 'included angle' mean in the context of the Law of Cosines?

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The included angle is the angle formed between two sides of a triangle. In the Law of Cosines, it is the angle opposite the side being calculated.

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How can the Law of Cosines be used to find the length of a side in a triangle?

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To find the length of a side using the Law of Cosines, rearrange the formula to isolate the side of interest. For example, to find side c: c = √(a² + b² - 2ab cos(C)).

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What is the relationship between the Law of Cosines and the Pythagorean theorem?

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The Law of Cosines generalizes the Pythagorean theorem. When the angle C is 90 degrees, cos(90°) = 0, and the Law of Cosines simplifies to the Pythagorean theorem: c² = a² + b².

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Can the Law of Cosines be used for non-right triangles?

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Yes, the Law of Cosines is applicable to all types of triangles, including acute and obtuse triangles.

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What is the significance of the cosine function in the Law of Cosines?

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The cosine function relates the angle of the triangle to the lengths of its sides. It helps to account for the angle's effect on the triangle's shape and side lengths.

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How do you determine which law to use when solving a triangle problem?

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Assess the given information: use the Law of Cosines for SAS or SSS cases, and the Law of Sines for ASA, AAS, or SSA cases.

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What is an example of using the Law of Cosines to solve for an angle?

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To find angle C, rearrange the Law of Cosines: cos(C) = (a² + b² - c²) / (2ab). Then use the inverse cosine function.