The key difference between the sine law and the cosine law lies in when they are applicable based on the information you have about a triangle.
Sine Law vs. Cosine Law: A Comparison
Here's a breakdown of when to use each law:
| Feature | Sine Law | Cosine Law |
|---|---|---|
| Applicable When Given: | Two angles and one side (AAS or ASA) Two sides and a non-included angle (SSA) |
Three sides (SSS) Two sides and the included angle (SAS) |
| Formula (Standard Triangle ABC): | a/sin(A) = b/sin(B) = c/sin(C) | a² = b² + c² - 2bc cos(A) b² = a² + c² - 2ac cos(B) c² = a² + b² - 2ab * cos(C) |
| What it Solves For: | Unknown sides or angles | Unknown sides or angles |
When to Use Each Law: More Detail
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Sine Law: If you know two angles and one side, you can use the sine law to find the remaining sides. Alternatively, if you know two sides and an angle that is not between them (SSA, also known as the ambiguous case), you can use the sine law to find the other angles. Be aware of the ambiguous case where there might be two possible solutions, one solution, or no solutions.
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Cosine Law: Use the cosine law when you know all three sides of a triangle (SSS) and want to find an angle. You can also use the cosine law when you know two sides and the angle between them (SAS) and want to find the third side.
Example Scenarios
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Example 1 (Sine Law): You have a triangle where angle A is 30 degrees, angle B is 60 degrees, and side a is 10 cm. You want to find side b. Since you have two angles and one side, you can use the sine law.
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Example 2 (Cosine Law): You have a triangle where side a is 5 cm, side b is 7 cm, and the angle between them (angle C) is 45 degrees. You want to find side c. Since you have two sides and the included angle, you use the cosine law.
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Example 3 (Cosine Law): You have a triangle where side a is 3 cm, side b is 4 cm and side c is 6 cm. You want to find angle A. Since you have three sides, you use the cosine law.
In summary, the sine law and cosine law are distinct tools used to solve for unknown sides and angles in triangles, selected based on the known information. Choosing the correct law depends on whether you have angle-side-angle (ASA), angle-angle-side (AAS), side-side-angle (SSA), side-angle-side (SAS), or side-side-side (SSS) information.
