Properties of Triangles

The sine, cosine and tangent rules

A triangle with sides of length a, b and c and angles of α, β and γ respectively.

The law of sines, or sine rule states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is

This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle with angles α, βand γ is similar to a triangle with side lengths equal to sinα, sinβ and sinγ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sinα, sinβand sinγ. The side whose length is sinα is opposite to the angle whose measure is α, etc.

The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side. As per the law:

For a triangle with length of sides a, b, c and angles of α, β, γ respectively, given two known lengths of a triangle a and b, and the angle between the two known sides γ (or the angle opposite to the unknown side c), to calculate the third side c, the following formula can be used:

The law of tangents or tangent rule, is less known than the other two. It states that:

Area Formulae

The height of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula Area = (1/2)bh derived above, the area of the triangle can be expressed as:

(where α is the interior angle at A, β is the interior angle at B, γ is the interior angle at C and c is the line AB). Projection formulae

b = c cos A + a cos C

c = a cos B + b cos A

The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to 180⁰. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

Triangles can be classified by the lengths of their sides or magnitude of their angles. Isosceles triangles have two equal sides and two congruent (equal) angles. Equilateral, or equiangular, triangles have three equal sides and angles. If no sides are equal, the triangle is a scalene triangle. All of the angles in an acute triangle are less than 90⁰ and at least one of the angles in an obtuse triangle is greater than 90⁰ . Triangles, such as these, which do not contain a 90⁰ angle, are generally known as oblique triangles. Right triangles, the most important ones to trigonometry, are those which contain one 90⁰angle.

Triangles which have proportional sides and congruent angles are called similar triangles. The concept of similar triangles, one of the basic insights in trigonometry, allows us to determine the length of a side of one triangle if we know the length of certain sides of the other triangle.

Sub-Multiple /(half) angle formulae