Law of Sines & Cosines

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What is the Law of Sines?

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The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. It can be expressed as: 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 expressed as: 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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The area (A) of a triangle can be found using the formula: A = (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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What is the formula for the area of a triangle using the Law of Cosines?

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The area can also be calculated using the formula: A = (1/2) * a * b * sin(C), derived from the Law of Cosines.

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How do you find an angle using the Law of Sines?

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To find an angle, rearrange the Law of Sines: sin(A)/a = sin(B)/b. Use inverse sine (sin⁻¹) to calculate the angle.

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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 opposite side. This relationship is crucial for applying the Law of Sines and Cosines.

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What is the significance of the included angle in the Law of Cosines?

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The included angle is the angle between the two sides being considered. It is essential for calculating the length of the third side.

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How can you determine if a triangle is acute, right, or obtuse using the Law of Cosines?

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By evaluating the cosine of the angles: if cos(C) > 0, the triangle is acute; if cos(C) = 0, it is right; if cos(C) < 0, it is obtuse.

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What is the formula for finding the length of a side using the Law of Cosines?

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To find the length of side c, use: c = √(a² + b² - 2ab*cos(C)).

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How do you apply the Law of Sines to find missing sides in a triangle?

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Use the ratio a/sin(A) = b/sin(B) to solve for the unknown side, ensuring you have at least one side and its opposite angle.