# Law of sines

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###### Topic Notes

In this section, we will learn about the Law of Sines, also known as the Sines Rule. The Law of Sines is a formula that models the relationship between the sides and the angles of any triangle, be it a right-angled triangle, an obtuse triangle, or an acute triangle. In order to use the Law of Sines, we need to satisfy the "one pair, one additional information" condition (i.e. Angle-Angle-Side abbreviated as AAS, and Angle-Side-Angle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines.

**Law of Sine**

For any $\triangle$

*ABC*,

$\frac{a}{\sin(A)}$ $=\frac{b}{\sin(B)}$ $=\frac{c}{\sin(C)}$

and,

$\frac{\sin(A)}{a}$ $=\frac{\sin(B)}{b}$ $=\frac{\sin(C)}{c}$

Use the Law of Sine when given a

*pair*!

**Ambiguous case**

Ambiguous case of the Law of Sine arises when given SSA (side-side-angle)

Step 1) Use the

*given angle*to find the height of the triangle: $h=b \sin (A)$

Step 2) Check if,

$Side\;a$ < $h$, then

*no triangles*

$Side\;a=h$, then

*1 triangle*

$Side\;a$ > $h$, then

*1 triangle*

$h$ < $Side\;a$ < $Side\;b$, then

*2 triangles*

Step 3) Solve the triangle(s)!

###### Basic Concepts

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