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Sine Rule

So far, we’ve seen the trigonometry for right-angled triangles. However, we can also apply the trigonometric rules for non-right angled triangles. Let’s first start by the naming conventions for a non-right angled triangle.

Each triangle has three points (written in capital letters) that also identify the angle they represent. The sides of a triangle are named with lower case letters opposite the angle they face. For example the side opposite to angle A is given as ‘a’, and so on.

There are two trigonometric rules for non-right angled triangles – sine rule and cosine rule. In this section, we look at the sine rule.


Definition of Sine Rule

The sine rule states that, in a non-right angled triangle,

{sin~A}/a  =  {sin~B}/b  =  {sin~C}/c is a constant.

sine ruleLet’s see how the above formula is arrived at. In the triangle ABC, draw a perpendicular to BC, so we get two right angled triangles, ABD and ADC.

From DeltaABD, sin B = h/c, hence

h  =  c sin B

From DeltaADC, sin C = h/b, hence

h  =  b sin C

Using the above 2 equations for h, we get h = c sin B = b sin C.

Therefore {sin B}/b  =  {sin C}/c

Similarly we can show that {sin B}/b  =  {sin A}/a.

Hence {sin A}/a  =  {sin B}/b  =  {sin C}/c

Remember: We use sine rule when we have opposite set of pairs of angles, and the sides are involved.


Let us look at some examples of sine rule:

Example 1: Find the unknown side x in the triangle ABC.example of sine rule-1

x/{sin60^circ}  =  7/{sin80^circ}

x  =  7 x {sin60^circ}/{sin80^circ}

=  6.1557 cm


Example 2: Find the unknown angle.example of sine rule-2

{sin30^circ}/8  =  {sinx^circ}/6

{sinx^circ} =  {6~*~sin30^circ}/8

=  3/8  =  0.375

x  =  sin^-1(0.375)

=  22.024°  =  22°1′ (to the nearest minute)


Summary of Sine Rule

  1. When you have to find unknown sides, use the sine rule formula –a/{sin A}  =  b/{sin B}  =  c/{sin C}
  2. When you have to find unknown angles, use the following sine rule formula –{sin A}/a  =  {sin B}/b  =  {sin C}/c