Formula for Sine Rule

Let a, b, and c be the lengths of the three sides of a triangle ABC and A, B, and C by their respective opposite angles. Now the expression for the Sine Rule is given as,

sin A/a = sin B/b = sin C/c = k

a/sin A = b/sin B = c/sin C = k

Sine Rule Proof

In triangle ABC, the sides of the triangle are given by AB = c, BC = a and AC = b.

 

Let us draw a perpendicular BD that is perpendicular to AC. Now we have two right-angles triangles ADC and BDC.

BD = h is the height of the triangle ABC. 

In triangle ADC,

sin A = h/c ⇢ (1)

In triangle BDC,

sin C = h/a ⇢ (2)

Now by dividing the equations (1) and (2).

We get, sin A/sin C = a/c ⇒ a/sin A = c/sin C ⇢ (3)

Similarly, draw a perpendicular AE that is perpendicular to BC. Now AEB and BEC are the right angled triangles separated by h_2.

In triangle AEB,

 sin B = H/c ⇢ (4)

In triangle AEC,

sin C = Hsolve/b ⇢ (5)

Now by dividing equations (4) and (5),

sin B/sin C = b/c ⇒ b/sin B = c/sin C ⇢ (6)

Now by equating the equations (3) and (6) we get,

a/sin A = b/sin B = c/sin C

sin A/a = sin B/b = sin C/c

Sine Rule which is also known as the Law of Sine, gives the relationships between sides and angles of any triangle. Sine Rule is a powerful tool in trigonometry that can be used to find solutions for triangles using various properties of triangles. With the help of the Sine rule, we can find any side of a triangle with ease only if we are given the angles and any of the sides of the triangle. In this article, we will explore all the aspects of the sine rule, including its formula, proof, and applications. Other than these all topics, we will also learn how to solve problems based on the sine rule or law of sine. So, let’s learn about the sine rule in this article.