Law of sines  Trigonometric Ratios and Angle Measures
Law of sines
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 rightangled 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. AngleAngleSide abbreviated as AAS, and AngleSideAngle abbreviated as ASA). We will also explore the concept of the Ambiguous Case of the Law of Sines.
Lessons
Notes:
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 (sidesideangle)
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)!

2.
Given the following triangle $\triangle ABC$,