Honors Geometry -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 is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides and the cosine of the included angle. Formula: c² = a² + b² - 2ab * cos(C).

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How do you find the length of side c using the Law of Cosines?

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To find side c, use the formula: c = √(a² + b² - 2ab * cos(C)), where a and b are the lengths of the other two sides and C is the included angle.

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What is the formula for finding angle C using the Law of Cosines?

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To find angle C, use the formula: C = cos⁻¹((a² + b² - c²) / (2ab)).

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If a = 5, b = 6, and C = 60°, what is the length of side c?

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c = √(5² + 6² - 2 * 5 * 6 * cos(60°) = √(25 + 36 - 30) = √31 ≈ 5.57.

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If a = 8, b = 10, and c = 12, what is the measure of angle C?

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C = cos⁻¹((8² + 10² - 12²) / (2 * 8 * 10)) = cos⁻¹((64 + 100 - 144) / 160) = cos⁻¹(20/160) = cos⁻¹(0.125) ≈ 82.82°.

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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 angle C is 90°, the Law of Cosines simplifies to the Pythagorean Theorem: c² = a² + b².

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How can the Law of Cosines be used to solve for an unknown side in a triangle?

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By rearranging the Law of Cosines formula, you can isolate the unknown side and calculate its length using the known sides and the included angle.

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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, allowing for the calculation of unknown sides or angles.

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If angle A = 45°, a = 7, and b = 9, how do you find angle B?

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Use the Law of Cosines to find side c first, then apply the Law of Sines to find angle B.

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What is the Law of Cosines used for in real-world applications?

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The Law of Cosines is used in various fields such as engineering, physics, and computer graphics to calculate distances and angles in non-right triangles.