The Law of Sines is a powerful relationship that connects every side of a triangle to the sine of its opposite angle. It works for any triangle, including scalene triangles where all sides and angles may be different. This makes it especially useful when right triangle trigonometry is not enough.
Surveying, navigation, architecture, and physics problems often depend on solving oblique triangles with this law.
The key idea is that each side divided by the sine of its opposite angle gives the same value throughout the triangle. This shared ratio lets you find missing sides or angles when you know enough matching side and angle information. The Law of Sines is commonly used in ASA, AAS, and some SSA cases.
The SSA case must be handled carefully because it can produce zero, one, or two possible triangles.
Key Facts
- Law of Sines: a/sin A = b/sin B = c/sin C
- Equivalent form: sin A/a = sin B/b = sin C/c
- Use the Law of Sines when you know ASA, AAS, or SSA information.
- Angles in any triangle add to 180 degrees: A + B + C = 180 degrees
- To find a side: a = b sin A/sin B, if angle A is opposite side a and angle B is opposite side b.
- SSA can be ambiguous because sin θ = sin(180 degrees - θ), so two different angles can have the same sine value.
Vocabulary
- Law of Sines
- A triangle rule stating that each side length divided by the sine of its opposite angle is the same for all three sides.
- Opposite side
- The side across from a given angle in a triangle.
- Scalene triangle
- A triangle with three different side lengths and three different angle measures.
- SSA case
- A triangle situation where two sides and a non-included angle are known.
- Ambiguous case
- A situation in the SSA case where the given information may form two possible triangles, one triangle, or no triangle.
Common Mistakes to Avoid
- Matching a side with the wrong angle is incorrect because the Law of Sines only uses opposite pairs, such as side a with angle A.
- Forgetting that triangle angles sum to 180 degrees leads to impossible answers, especially when finding a third angle before using the formula.
- Assuming every SSA problem has one solution is wrong because the ambiguous case can create two triangles, one triangle, or no triangle.
- Rounding too early can change the final side length or angle noticeably, so keep several decimal places until the last step.
Practice Questions
- 1 In triangle ABC, A = 40 degrees, B = 65 degrees, and a = 12 cm. Find side b to the nearest tenth.
- 2 In triangle ABC, A = 52 degrees, a = 18 m, and b = 22 m. Use the Law of Sines to find possible values of angle B, then decide how many triangles are possible.
- 3 A student says that if a/sin A = b/sin B, then side a must always be the longest side. Explain why this reasoning is wrong and state what must be true for a to be the longest side.