Law of Sine and Law of Cosine

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What is the Law of Sines?

Back

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. It can be expressed as: a/sin(A) = b/sin(B) = 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 expressed as: c² = a² + b² - 2ab*cos(C).

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

Back

To find an unknown side using the Law of Sines, you can use the formula: a/sin(A) = b/sin(B), where 'a' and 'b' are the sides opposite angles A and B, respectively.

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

Back

To find an unknown angle using the Law of Sines, rearrange the formula: sin(A)/a = sin(B)/b, and solve for the angle using the inverse sine function.

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

Back

In any triangle, the larger the angle, the longer the side opposite that angle. Conversely, the smaller the angle, the shorter the side opposite it.

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When is it appropriate to use the Law of Cosines?

Back

The Law of Cosines is appropriate to use when you know two sides and the included angle (SAS) or all three sides (SSS) of a triangle.

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What is the formula for finding the area of a triangle using the Law of Sines?

Back

The area of a triangle can be found using the formula: Area = (1/2) * a * b * sin(C), where 'a' and 'b' are two sides and 'C' is the included angle.

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What is the significance of the angle opposite the longest side in a triangle?

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The angle opposite the longest side in a triangle is the largest angle, and it is crucial for determining the triangle's properties.

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How can the Law of Cosines be derived from the Pythagorean theorem?

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The Law of Cosines can be derived from the Pythagorean theorem by considering a triangle and applying the theorem to the components of the triangle formed by dropping a perpendicular from one vertex to the opposite side.

Front

What is the relationship between the sine and cosine of complementary angles?

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The sine of an angle is equal to the cosine of its complementary angle: sin(A) = cos(90° - A).