The Law of Sines or Sine Formula
In each case, we have \( \frac{a}{\sin A}=2 R \)
3.7.1 Law of Sines The Law of Sines is a relationship between the angles and the sides of a triangle. Whilk, Thus, \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R \).
Similarly, by considering angles \( B \) and \( C \), we can prove tha respectively.
(i) To find an angle if two sides and one angle which is not included, by them are given
(ii) To find a side, if two angles and one side which is opposite to one of given angles, in
Theorem 3.1 (Law of Sines): In any triangle, the lengths of the sides are proportional tofef
the opposite angles. That is, in \( \triangle A B C, \mid \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2 R \) where \( R \) is the ciliog (ii) The Law of Sines says that the sides of a trians of the triangle.
Proof. The angle \( A \) of the \( \triangle A B C \) is either acute or right or obtuse. Let \( O \) be the corl opposite angles.
(iii) Using the Law of Sines, it is impossible to find circumcircle of \( \triangle A B C \) and \( R \), its radius. \( -X \) and the included angle.
(iv) An interesting geometric consequence of the triangle is opposite to the largest angle. (Pro
Napier's Formula Theorem 3.2: In \( \triangle A B C \), we have (i) \( \tan \frac{A-B}{2}=\frac{a-b}{a+b} \cot \frac{C}{2} \) (ii) \( \tan \frac{b-C}{b-c} \cot \frac{A}{2} \) Thg Quad Camera-a h my Galaxy A12A is acute
Figure \( 3.18 \)
