4. Other types of triangles

So far, we have looked at trigonometric functions in the context of a right-angled triangle, where one of the angles is 90°. There are other types of triangles:

• Equilateral triangles have all three sides of equal length, and all angles are 60°.
• Isosceles triangles have two sides of equal length, so that two angles are also equal.
• Scalene triangles have no sides or angles equal

For all triangles:

• The sum of the three angles of a triangle is always 180°.
• The longest side is opposite to the largest angle.
• A triangle can only have one angle that is equal to 90° (a right angle) or greater (an obtuse angle).

There are two important trigonometric rules that apply to all triangles. First, look at this scalene triangle with sides a, b and c opposite angles α, β and γ respectively:

The sine rule states that:

a/sinα = b/sinβ = c/sinγ

This means that all of the sides and angles of any triangle can be calculated if one angle and its opposite side, plus one other angle or side are known.

The cosine rule states that:

a2 = b2 + c2 – 2bc cosα
b2 = a2 + c2 – 2ac cosβ
c2 = a2 + b2 – 2ab cosγ

This allows the length of one side of a triangle to be calculated if the lengths of the other two sides and the size of the included angle are known, or any angle if the lengths of all three sides are known.