Law of sines

Definition

The law of sines states that in any triangle, the ratios of the lengths of a side to the sine of its opposite angle are equal. This relationship helps solve for unknown sides and angles in non-right triangles, making it an essential tool in trigonometry and geometry, especially when working with oblique triangles.

5 Must Know Facts For Your Next Test

1. The law of sines is written as $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$, where 'a', 'b', and 'c' are the sides of the triangle and 'A', 'B', and 'C' are the angles opposite those sides.
2. It can be used to find missing angles or sides when you have either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
3. The law of sines is especially useful for solving triangles that are not right-angled, providing a way to apply trigonometric principles beyond right triangles.
4. When using the law of sines, the ambiguous case can arise in SSA situations, leading to either zero, one, or two possible triangles.
5. The law of sines also highlights the relationship between the angles and sides of a triangle, showing how they are proportional to each other through the sine function.

Review Questions

• How does the law of sines apply when solving for unknown sides and angles in a triangle? Provide an example.
  • The law of sines applies by allowing you to set up ratios between the sides and their opposite angles. For example, if you know two angles and one side of a triangle (AAS), you can use the law of sines to find the other sides. If angle A is 30 degrees, angle B is 60 degrees, and side a is 10 units long, you can set up the ratio $$\frac{10}{\sin 30} = \frac{b}{\sin 60}$$ to solve for side b.
• Discuss the ambiguous case of the law of sines. Under what conditions does this occur?
  • The ambiguous case occurs when applying the law of sines in an SSA configuration, where you have two sides and a non-included angle. In this situation, there may be zero, one, or two valid triangles that satisfy those measurements. For instance, if one side is shorter than the height from the opposite angle, no triangle can be formed. If it equals the height, exactly one triangle exists. If it is greater than the height but less than the longer side, two different triangles can be constructed.
• Evaluate how understanding the law of sines enhances problem-solving abilities in trigonometry beyond just right triangles.
  • Understanding the law of sines expands problem-solving capabilities by enabling you to tackle a broader range of problems involving non-right triangles. It provides a systematic approach for finding unknown angles and sides in various scenarios where right triangle relationships do not apply. Mastery of this law allows students to engage with real-world applications such as navigation, architecture, and physics, where oblique triangles frequently arise.

Related terms

Triangle: A polygon with three edges and three vertices, which can be classified based on side lengths or angles.

Oblique Triangle: A triangle that does not contain a right angle, which can be either acute or obtuse.

Sine Function: A mathematical function defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.