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.
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
= 22.024° = 22°1′ (to the nearest minute)
Summary of Sine Rule
- When you have to find unknown sides, use the sine rule formula – = =
- When you have to find unknown angles, use the following sine rule formula – = =