PreCalculus- Law of Sines and Cosines

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What is the Law of Sines?

Back

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: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

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What is the Law of Cosines?

Back

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is given by: \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

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How do you determine if a triangle can be formed with given sides?

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Use the triangle inequality theorem: The sum of the lengths of any two sides must be greater than the length of the third side.

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

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The area \( A \) of a triangle can be calculated using the formula: \( A = \frac{1}{2}ab \sin(C) \)

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What is the significance of the angle of elevation in real-world problems?

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The angle of elevation is the angle between the horizontal line and the line of sight to an object above the horizontal. It is used in problems involving heights and distances.

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How do you solve for an unknown side using the Law of Sines?

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To find an unknown side, rearrange the Law of Sines: \( a = \frac{b \sin(A)}{\sin(B)} \)

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What is the ambiguous case in the Law of Sines?

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The ambiguous case occurs when two sides and a non-included angle (SSA) are known, leading to zero, one, or two possible triangles.

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How do you find the distance between two points using the Law of Cosines?

Back

Use the Law of Cosines to find the distance \( c \) between two points A and B: \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

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

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In a triangle, larger angles are opposite longer sides, and smaller angles are opposite shorter sides.

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How do you round answers in triangle problems?

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Round answers to the nearest hundredth or as specified in the problem statement.