The Law of Sines is a rule that connects each side of a triangle to the sine of its opposite angle. It works for any triangle, not just right triangles, which makes it useful when basic right triangle trigonometry is not enough. Students use it to find missing side lengths and angle measures when enough information about a triangle is known.
It is especially helpful in geometry, navigation, surveying, physics, and engineering.
The key idea is that larger angles face longer sides, and the ratios a/sin A, b/sin B, and c/sin C are all equal in the same triangle. To use the rule correctly, each side must be paired with the angle directly across from it. The Law of Sines is most direct for ASA, AAS, and some SSA triangle information.
The SSA case can be ambiguous because the same given information may create 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
- Angles in a triangle: A + B + C = 180°
- Use the Law of Sines when you know an angle and its opposite side, plus one other side or angle.
- ASA and AAS usually give one triangle when the measurements are valid.
- SSA can give zero, one, or two triangles because sin θ = sin(180° - θ).
Vocabulary
- Law of Sines
- A trigonometric relationship stating that the ratio of a side length to the sine of its opposite angle is the same for all three sides of a triangle.
- Opposite Side
- The side directly across from a given angle in a triangle.
- Included Angle
- An angle formed between two known sides of a triangle.
- SSA Case
- A triangle setup with two sides and a non-included angle known, which may lead to zero, one, or two possible triangles.
- Ambiguous Case
- A situation in the SSA case where the given measurements can produce more than one valid triangle.
Common Mistakes to Avoid
- Pairing a side with the wrong angle. The Law of Sines only works when each side is matched with the angle directly opposite it.
- Forgetting that triangle angles sum to 180°. Always find the third angle with A + B + C = 180° before solving a missing side when two angles are known.
- Using the inverse sine result as the only possible angle in SSA. Since sin θ = sin(180° - θ), a second angle may also create a valid triangle.
- Rounding too early in a multi-step problem. Keep several decimal places until the final answer to avoid noticeable error in side lengths or angles.
Practice Questions
- 1 In triangle ABC, A = 42°, B = 68°, and a = 12 cm. Find angle C and side b to the nearest tenth.
- 2 In triangle ABC, A = 35°, a = 9 m, and b = 13 m. Use the Law of Sines to determine whether there are zero, one, or two possible triangles, and find the possible value or values of angle B.
- 3 A student is given two sides and a non-included angle and immediately draws one triangle as the only solution. Explain why this reasoning may be incomplete and describe how the Law of Sines helps check the possibilities.