Law of Sines and Cosines

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What is the Law of Sines?

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The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all three sides of the triangle. It is expressed as: a/sin(A) = b/sin(B) = c/sin(C).

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When do you use the Law of Sines?

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The Law of Sines is used when you have either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) in a triangle.

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What is the Law of Cosines?

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The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is expressed as: c² = a² + b² - 2ab*cos(C).

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When do you use the Law of Cosines?

Back

The Law of Cosines is used when you have either two sides and the included angle (SAS) or all three sides (SSS) of a triangle.

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What is the significance of the angle in the Law of Sines?

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The angle in the Law of Sines is crucial as it helps determine the relationship between the sides of the triangle and allows for the calculation of unknown angles or sides.

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How do you determine the number of unique triangles using the Law of Sines?

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To determine the number of unique triangles, analyze the given information (angles and sides) and apply the ambiguous case of the Law of Sines. If the conditions do not satisfy triangle inequality, there may be no triangle.

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What is the triangle inequality theorem?

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The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

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Example: Given A=37°, a=8, b=14, how many triangles can be formed?

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No triangles can be formed because the conditions do not satisfy the triangle inequality.

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Example: Given J=98°, j=29, p=6, how many triangles can be formed?

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1 triangle can be formed as the conditions satisfy the triangle inequality.

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What is the formula to find the area of a triangle using the Law of Sines?

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Area = (1/2) * a * b * sin(C), where a and b are two sides and C is the included angle.