Mathematics, queen of science is a such a subject that has got the number of people hating it more rather than falling in love with it. This subject is the basic foundation for most of the other subjects like mathematical physics, fluid mechanics, probability, electrical engineering and mathematical statistics.

But there are few branches in mathematics which are fun to solve. Again it all depends on the individual. Arithmetic, algebra, geometry, trigonometry are few branches of mathematics.

**Triangle:**

A triangle is the polygon which has the three edges and the three vertices which is one of the basic shapes in geometry. There are different types of triangles depending upon lengths and internal angles. Equilateral, isosceles and scalene are based on lengths of sides and acute, obtuse and right-angled triangles are based on internal angles.

There are few conditions for any triangle to exist. They are given as:

- Condition on sides: According to this condition sum of the length of any two sides of the triangle must be greater than or equal to the length of the third side. We call this one as triangle-inequality.
- Condition on the angles: According to this condition, each angle of the triangle should be positive and the sum of all the angles must sum up to { 180 }^{ 0 } .
- Trigonometric conditions: There are a few trigonometric formulas that are used to define the existence of any triangle.

**Trigonometry:**

The trigonometry is defined as one of the branches of the mathematics that is used to study the relationship between lengths and angles of the triangle. The trigonometric functions are used to find the ratios of the sides of the triangle irrespective of the overall size of the triangle.

Using the same trigonometric functions, trigonometric formulas are derived. Following are the important trigonometric functions.

- Sine function(sin) : The ratio of opposite side angle to the hypotenuse.
- Cosine function(cos) : The ratio of the adjacent side angle to the hypotenuse.
- Tangent function(tan) : It is the ratio of opposite side to the adjacent side.

Reciprocals of these functions are also used in trigonometric formulas and they are:

- Cosecant function(cosec): It is the ratio of the hypotenuse to opposite.
- Secant function(sec): The ratio of the hypotenuse to adjacent.
- Cotangent function(cot): It is the ratio of adjacent to opposite.

Trigonometry also finds the applications in scientific fields like probability theory, electrical engineering, and statistics and also in solving Fourier transforms Fourier series.

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