Sine Rule Formula

The Law of Sine is also known as Sine Formula or Sine Rule in Trigonometry. These rules deal with sides of a triangle with any of its angles. Let’s look at the Sine rule formula.

Sine Rule Formula

\(\begin{array}{l}\frac{a}{sin A}\end{array} \)
=
\(\begin{array}{l}\frac{b}{sin B}\end{array} \)
=
\(\begin{array}{l}\frac{c}{Sin C}\end{array} \)

Where a, b, c are the lengths of the sides opp to angle A, angle B, and Angle C of the triangle for which we will use the Law of sines.

Law of Sines Formula Example

Example: If  angle B = 210, angle C= 460 and the side AB = 9 cm in a triangle is given. Find the other sides of triangle.

Solution:

Given: two angles and a side

Let’s use the Sine rule to solve this.

As the sum of angles in a triangle is 1800

Accordingly, angle A = 1130

As AB = c = 9 cm.

Use the Sine Rule:

\(\begin{array}{l}\large \frac{a}{sin 113^{\circ}}= \frac{b}{sin 21^{\circ}}= \frac{9}{Sin 46^{\circ}}\end{array} \)
\(\begin{array}{l}\large \frac{b}{sin 21^{\circ}} = \frac{9}{sin 46^{\circ}}\end{array} \)

b = sin 210 x 9 /sin 460

= 4.484 cm

a = sin 1130 x 9 /sin 460

=11.517 cm.

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