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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.

Radius of Circumcircle

Consider a triangle ΔABC incscrbed in a circle with centre O and radius R, the ratios of the side opposite to an angle to the corresponding sine function of the angle are equal to 2R. Similarly, there are three cases to be considered in proving the ratios are equal to the diameter of the circumcircle for the three different types of angled triangles. The acute angled triangle with all the internal angles are less than π/2. The right angled triangle with one of the internal angles is equal to π/2. The obtuse angled triangle with one of the internal angles is greater than π/2 but less than π.

For an acute angled triangle ΔABC incscrbed in a circle with centre O and radius R, three isosceles triangles can be constructed by joining the vertices, A, B, and C to the intersection of the three perpendicular bisectors of the triangle, O with the altitude of each isosceles triangle coninciding with the perpendicular bisector.  Imply

Consider isosceles triangle ΔCOB, imply

For an obtuse angled triangle ΔABC incscrbed in a circle with centre O and radius R, three isosceles triangles can be constructed by joining the vertices, A, B, and C to the intersection of the three perpendicular bisectors of the triangle, O with the altitude of each isosceles triangle coninciding with the perpendicular bisector.  Imply

Consider isosceles triangle ΔCOB, imply

For an right angled triangle ΔABC incscrbed in a circle with centre O and radius R, two isosceles triangles can be constructed by joining the vertices, B to the intersection of the three perpendicular bisectors of the triangle, O with the altitude of each isosceles triangle coninciding with the perpendicular bisector.  Imply

Consider isosceles triangle ΔCOB, imply