Everything that we see around is somewhere tangled with mathematics. Mathematics is the language, through which this creation around the world, communicates. To know better, how every single thing works around the world or this cosmos, we need to learn the language of mathematics. This language has several different chapters. Each and every chapter are interrelated. You may name these chapters as arithmetic, algebra, geometry, trigonometry, Calculous and so on. So when you complete all the chapters one by one, you can see how beautifully everything is communicating with everything.
Arithmetic was basic and invented first. Everything related to counting finite numbers was taken care of by arithmetic. And when some of the parameters were known, some of them were unknown, Algebra came into action. People were measuring line, length and perimeters using geometry but when the question of Triangle with different angles came, Trigonometry was invented.
What Is Trigonometry?
Trigonometry is that important chapter of mathematics that deals with triangles. To be more specific, a triangle that is in a plane and one angle of that triangle is 90 Degree. In particular the ratio and relationship between Triangles sides and angles.
Why Do We Need It?
All the branches of engineering, starting from civil to computer science, are there because trigonometry was invented. Whenever we needed applied mathematics, Trigonometry was the first option. From astronomy to Music Theory, Trigonometry formulas have a strong presence all over them. Experts from the Vedantu Trigonometry formula department helped us with quick tips for the students to crack it in a easy way.
How Do We Do It – The Trigonometry Formula
There are six functions that are the core of trigonometry. Among them, Three functions are the primary ones. Those are –
- Sine (Sin)
- Cosine (Cos)
- Tangent (Tan)
The other three are not used much often and if they are needed, can be derived from the primary functions. Those are –
- Secant (Sec)
- Cosecant (Cosec)
- Cotangent (Cot)
Definitions of the six functions –
Consider the right triangle on the left. For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle. The only difference between the six functions is which pair of sides we use.
In the following table
a is the length of the side adjacent to the angle (x) in question.
o is the length of the side opposite the angle.
h is the length of the hypotenuse.
“x” represents the measure of their angle in either degrees or radians.
| Sine | Sin x = o/h | Primary | |
| Cosine | Cos x = a/h | Primary | |
| Tangent | Tan X = o/a | Primary | |
| Secant | Sec x = h/a | Derived | Sec x = 1/Cos x |
| Cosecant | Cosec x = h/o | Derived | Cosec x = 1/Sin x |
| Cotangent | Cot x = a/o | Derived | Cot x = 1/Tan x |
For example, in the figure above, the cosine of x is the side adjacent to x (labelled a), over the hypotenuse (labelled h):
Cos x = a/h
If a=12cm,
and h=24cm,
then cos x = 0.5
Inverse Functions
For each of the six functions, there is an inverse function that works in reverse. The inverse function has the letters ‘ARC' in front of it.
For example, the inverse function of COS is ARCCOS. While COS tells you the cosine of an angle, ARCCOS tells you what angle has a given cosine.
Trigonometry Functions Of Large and/or Negative Angles
The six functions can also be defined in a rectangular coordinate system. This allows them to go beyond right triangles, to where the angles can have any measure, even beyond 360°, and can be both positive and negative.
Identities – Replacing A Function With Others
Trigonometric identities are simply ways of writing one function using others. For example, from the table above we see that
Sec x = 1/Cos x
This equivalence is called an identity. If we had an equation with sec x in it, we could replace sec x with one over cos x if that helps us reach our goals. There are many such identities.
Not Just Right Triangles
These Trigonometry Formulas are defined using a right triangle but used in other triangles too. For example, the Law of Sines and the Law of Cosines can be used to solve any triangle – not just right triangles
Function Graphs
The functions can be graphed, and some, notably the SIN function, produce shapes that frequently occur in nature. Examples see the graph of the SIN function, often called a sine wave above.
This is all in a nutshell offered for you to have a basic understanding of how trigonometry works. But this is just the beginning. We have a sea to explore here. Step by step.
