Law of Sines

Definition

The Law of Sines is a trigonometric equation that relates the lengths of the sides of a triangle to the sines of its angles. It states that for any triangle, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, where $a$, $b$, and $c$ are the lengths of the sides opposite angles $A$, $B$, and $C$ respectively.

5 Must Know Facts For Your Next Test

1. The Law of Sines can be used to solve for unknown sides and angles in any triangle, not just right triangles.
2. It is particularly useful in cases of oblique triangles (triangles without a right angle).
3. The Law of Sines can lead to ambiguous cases (SSA condition) where two different triangles may satisfy the given conditions.
4. To apply the Law of Sines, at least one angle-side opposite pair must be known.
5. The formula can be rearranged to solve for an unknown angle: $\sin A = \frac{a \cdot \sin B}{b}$.

Review Questions

• How can you use the Law of Sines to find an unknown side in a triangle?
• What are ambiguous cases in the context of the Law of Sines?
• When applying the Law of Sines, what minimal information must you have about a triangle?

Related terms

Law of Cosines:

A trigonometric identity that relates the lengths of all three sides of a triangle to one cosine angle: $c^2 = a^2 + b^2 - 2ab\cos(C)$.

Oblique Triangle: A triangle that does not contain a right angle.

Ambiguous Case: A situation in which two different triangles can be formed with given SSA conditions when using the Law of Sines.