Law of Sines & Cosines

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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.

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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 used to find a side or angle in a triangle when the Law of Sines cannot be applied.

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When do you use the Law of Sines?

Back

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?

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

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What is the formula for the Law of Sines?

Back

The formula is: (a/sin(A)) = (b/sin(B)) = (c/sin(C)), where a, b, and c are the lengths of the sides and A, B, and C are the angles.

Front

What is the formula for the Law of Cosines?

Back

The formula is: c² = a² + b² - 2ab * cos(C), where a, b, and c are the lengths of the sides and C is the angle opposite side c.

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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 formula to isolate the sine of the angle, then use the inverse sine function.

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How do you find a side using the Law of Sines?

Back

To find a side, rearrange the Law of Sines formula to isolate the side length, then multiply by the sine of the corresponding angle.

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

Back

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

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If angle A = 49° and angle B = 58°, what is angle C?

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Angle C can be found using the triangle sum theorem: C = 180° - (A + B) = 180° - (49° + 58°) = 73°.