Law of sines

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 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)}
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,
SideaSide\;a < hh,
then no triangles
then 1 triangle
SideaSide\;a > hh,
then 1 triangle
hh < SideaSide\;a < SidebSide\;b,
then 2 triangles

Step 3) Solve the triangle(s)!
  • 1.

  • 2.
    Given the following triangle ABC\triangle ABC,
    Using law of sines to find angles and side lengths of triangles
    Solve forC\angle C

    Solve for aa

  • 3.
    Solve for side xx
    law of sines and side lengths of triangles

  • 4.
    Ambiguous case: SSA triangles
    In DEF\triangle DEF, DE=21cmDE=21cm, \angle F=45F=45°, and EF=24cmEF=24cm; find DFDF.