Trigonometry

Trigonometric functions are related with the properties of triangles. Some laws and formulas are also derived to tackle the problems related to triangles, not just right-angled triangles.

The Law of Tangents

The law of Tangents or tangent rule are related to the tangent or cotangent functions of angles included in a triangle and the sides of the triangle. The Law of tangents states that for a given arbitrary triangle with angle-side opposite pairs (A,a), (B,b) and (C,c), the ratio of the difference of two sides to the sum of the two sides are equal to the ratio of the tangent function of the half of the difference of angles opposite to the two sides to the tangent function of the half of the sum of angles opposite to the two sides. Imply

Through rearrangement of terms, the law of tangents can be used in any case when two angles and one of their opposite sides are known. Imply

The law of tangents can be used in any case when two sides and the included angle are known. Since the internal angle sum of a triangle is equal to π, imply π=A+B+C and (A+B)=π-C . If C is know, then A+B is known and angles A and B can be deduced from A-B also. That is A=((A+B)+(A-B))/2. Imply

The law of tagents can be derived from the law of sines trigonometrically. According to the law of sines, the ratio of side over the angle of the side opposite is equal to a constant k, sides of a triangle can therefore be expressed in terms of the sine function of the angle of the side opposite. That is a=k(sin A), b=k(sin B), and c=k(sin C).  The ratio of the difference of two sides to the sum of the two sides for any triangly can also be obtained accordingly and is equal to the ratio of the difference of the two sine functions of the two angles of sides opposite to the sum of the  two sine functions of the two angles of sides opposite. And the ratio can further be reduced to the law of tangent by using the trigonometryic identities. Imply