**Law of sines :**

The sides of a triangle are proportional to the sides of angles opposite to them is a triangle ABC,

a/sinA = b /sin B = c/sinC

**Case 1 :**

When triangle ABC is an acute angles triangle.

Draw AD perpendicular from A to the opposite side BC meeting it in the point D.

In the triangle ABD, we have

Sin B = AD/AB ==> sin B = AD/c ==> AD = c sin B ---(i)

In the triangle ACD, we have

Sin C = AD/AC ==> sin C = AD/b ==> AD = b sin C---(ii)

From (i) and (ii), we get

b/sin B = c/sin C

In a similar manner, by drawing a perpendicular from B on AC, we obtain

a/sin A = c/sin C

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

**Case II :**

When triangel ABC is an obtuse triangle

Draw AD perpendicular from A on CB produced meeting it in D.

sin ∠ABD = AD/AB

sin (180 – B) = AD/c

sin B = AD/c

AD = c sin B

In triangle ACD, we have

sin C = AD/AC

sin C = AD/b

AD = b sin C

From (i) and (ii), we obtain

c sin B = b sin C

b/sin B = c/sin C

Similarly, by drawing perpendicular from B on AC, we obtain

a/sin A = c/sin C

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

**Case III**

When triangle ABC is a right angled triangle.

In triangle ABC, we have

sin C = sin ∏/2 = 1 ==> sin A = BC/AB = a/c,

sin B = AC/AB = b/c ==> a/sin A = b/sin B = c

a/sin A = b/sin B = c/1

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

Hence, in all cases, we obtain

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

The above rules may also be expressed as

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

The sin rule is a very useful tool to express sides of a triangle in terms of the sines of angles and vice versa in the following manner.

Let a/sin A = b/sin B = c/sin C = k (say)

Then a = k sin A, b = k sin B, c = k sin C

Similarly,

sin A/a = sin B/b = sin C/c = λ (say)

sin A = λ a, sin B = λ b and sin C = λ c.

Some more results based on sin rule

In triangle ABC, we have

sin (B + C) = sin A sin (C + A) = sin B sin (A + B) = sin C |
cos (B + C) = -cos A, cos (C + A) = cos B, cos (A + B) = -cos C |

After having gone through the stuff given above, we hope that the students would have understood "Law of sines".

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