PreCalculus: Law of Sines and Law of 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 and angles in the triangle. Formula: a/sin(A) = b/sin(B) = c/sin(C).

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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 useful for finding a side or angle in a triangle when the Law of Sines cannot be applied. Formula: c² = a² + b² - 2ab*cos(C).

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How do you 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 the lengths of two sides and C is the included angle.

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How do you find the area of a triangle using the Law of Cosines?

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First, find the length of one side using the Law of Cosines, then use the area formula: Area = (1/2) * a * b * sin(C).

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What is the relationship between the angles and sides in a triangle?

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In any triangle, the larger the angle, the longer the side opposite that angle.

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How do you determine the smallest angle in a triangle?

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The smallest angle is opposite the shortest side of the triangle.

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

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

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What is the formula for finding an angle using the Law of Sines?

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To find angle A: A = sin⁻¹((a * sin(B)) / b).

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When should you use the Law of Sines?

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Use the Law of Sines when you have two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).

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When should you use the Law of Cosines?

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Use the Law of Cosines when you have two sides and the included angle (SAS) or all three sides (SSS).