Law of Sine and 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. Formula: a/sin(A) = b/sin(B) = c/sin(C).

Front

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 useful for finding an unknown side or angle in a triangle. Formula: c² = a² + b² - 2ab*cos(C).

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When do you use the Law of Sines?

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

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When do you use the Law of Cosines?

Back

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

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

Back

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

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

Back

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

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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 opposite side. Conversely, the smaller the angle, the shorter the opposite side.

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What is the significance of the sine function in triangles?

Back

The sine function relates the angle of a triangle to the ratio of the length of the opposite side to the hypotenuse in a right triangle.

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How can you determine if a triangle is valid using side lengths?

Back

A triangle is valid if the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality Theorem).

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What is the formula for the area of a triangle given three sides?

Back

Use Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2.