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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, = = is a constant. Let’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 ABD, sin B = , hence

h  =  c sin B

From ADC, sin C = , hence

h  =  b sin C

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

Therefore = Similarly we can show that = .

Hence = = 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.  = x  =  7 x =  6.1557 cm

Example 2: Find the unknown angle.  =  = = =  0.375

x  = =  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 – = = 2. When you have to find unknown angles, use the following sine rule formula – = = 