Law of sines

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,

asin⁡(A)\frac{a}{\sin(A)} =bsin⁡(B)=\frac{b}{\sin(B)} =csin⁡(C)=\frac{c}{\sin(C)}
and,
sin⁡(A)a\frac{\sin(A)}{a} =sin⁡(B)b=\frac{\sin(B)}{b} =sin⁡(C)c=\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=bsin⁡(A)h=b \sin (A)

Step 2) Check if,
Side  aSide\;a < hh,
then no triangles
Side  a=hSide\;a=h,
then 1 triangle
Side  aSide\;a > hh,
then 1 triangle
hh < Side  aSide\;a < Side  bSide\;b,
then 2 triangles

Step 3) Solve the triangle(s)!
Basic Concepts
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