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  1. Given the following triangle ABC\triangle ABC,
    Using law of sines to find angles and side lengths of triangles
    1. Solve forC\angle C
    2. Solve for aa
  2. Solve for side xx
    law of sines and side lengths of triangles
    1. Ambiguous case: SSA triangles
      In DEF\triangle DEF, DE=21cmDE=21cm, \angle F=45F=45°, and EF=24cmEF=24cm; find DFDF.
      Topic Notes
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      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)!