Question

What is trigonometry?

Answer 1

Trigonometry
Introduction to Trigonometry
Trigonometry is defined as one of the branches of mathematics that deals with the relationships that involve lengths and also the angles of triangles. In a simple manner, we can say that Trigonometry is the study of triangles. The word trigonometry was derived from the Greek word, where, ‘TRI‘ means Three, ‘GON‘ means sides and whereas the ‘METRON’ ways to measure. This concept is given to us by a Greek mathematician Hipparchus.
To be more specific, trigonometry is all about a right angled triangle, where one of the internal angles measure about ${90}^{\circ }$. Moreover, it is one of those divisions in mathematics that helps in finding the angles and missing sides in a triangle. In trigonometry, the angles are either measured in radians or degrees.
This branch divides into two sub-branches called as plane trigonometry and the spherical geometry. Trigonometry, in general, is about the trigonometric formulas, trigonometric ratios, and functions, Right-Angled Triangles, etc. Let us study all these topics in detail.
Introduction
Triangles
Triangle Inequality Theorem
Sine, Cosine and Tangent
Sine, Cosine and Tangent
Graphic Representation Of Inverse Trigonometric Function
Inverse Trigonometric Functions
Congruent and Similar
Congruence of Triangles
Similar Triangles
Theorems about Similar Triangles
Solving Triangles
Exterior Angle Theorem
Law of Sines
Law of Cosines
Area
Heron’s Fomula
Trigonometric Functions and Ratios:
The trigonometric ratios of a triangle are also called the trigonometric functions. There are three essential trigonometric functions in trigonometry known as Sine, cosine, and tangent and abbreviated as Sin, Cos, and Tan. How are these ratios or functions, evaluated in the case of a right-angled triangle?

Consider the right-angled triangle, where the longest side is called the hypotenuse, and the sides opposite to the hypotenuse is referred to as the adjacent and opposite. The trigonometric functions or the ratios for this triangle is calculated by the below formulas.
Sine ratio or Function is given as, $sin\theta =\frac{Opposite}{Hypotenuse}$
Tangent ratio or Function is given as, $tan\theta =\frac{Opposite}{Adjacent}$
Cosine ratio or Function is given as, $cos\theta =\frac{Adjacent}{Hypotenuse}$
Similar to the ratios sine, cosine and tangent, there are other three trigonometric ratios or functions in trigonometry called Cotangent, Cosecant, and Secant. The values of these trigonometric functions are evaluated by using the following formulas
Cosecant Function- $cosec\theta =\frac{Hypotenuse}{Opposite}$
Cotangent Function- $cot\theta =\frac{Adjacent}{Opposite}$
Secant Function – $sec\theta =\frac{Hypotenuse}{Adjacent}$

Inverse Trigonometric Ratios:
The inverse trigonometric functions are those functions which involve the inverses of cosine, tangent, and Sine. These opposites, are called as inverse trigonometric functions. By considering the right-angled triangle, the inverse functions, are given below-
$\text{cosec}\theta =\frac{1}{\sin \theta }=\frac{Hypotenuse}{opposite}\cot \theta =\frac{1}{\tan \theta }=\frac{adjacent}{opposite}\sec \theta =\frac{1}{\cos \theta }=\frac{hypotenuse}{adjacent}$

Trigonometric Ratios of Specific Angles:
Suppose if you are given the question – In a right-angled triangle $ABC$, if one side of the triangle is ${45}^{\circ }$, then what is the value of other side or angle?
For such questions, the below table will help you out in finding the trigonometric and inverse trigonometric ratios of different Angles of a triangle.

Pythagoras Theorem:
The Pythagoras theorem helps to know the relationship between the trigonometric identities, which, is discussed in the next sub-heading.

Pythagoras theorem states “ the square of the hypotenuse (c )equals to the sum of the squares of the adjacent( b ) and opposite ( a).”
In equation form, it is, given as: $c^2=a^2+b^2$

Trigonometric Identities:
An identity is a form of equation true for all the values of the variables, which both the sides of an equation is defined. These variables are either in the shape of a statement or even specified. The Pythagoras theorem is also one of the trigonometric identities. The Trigonometric Identities are the equations which are true in the case of Right Angled Triangles.
Some of the other identities or rather say formulas in trigonometry, are as given below-
1. Pythagorean Identities-
${\sin }^2\theta +{\cos }^2\theta =1{\tan }^2\theta +1={\sec }^2\theta {\cot }^2\theta +1={\text{cosec}}^2\theta \sin 2\theta =2\sin \theta \cos \theta \cos 2\theta ={\cos }^2\theta -{\sin }^2\theta \tan 2\theta =2\tan \frac{\theta }{(1-{\tan }^2\theta )}\cot 2\theta =\frac{{\cot }^2\theta -1}{2\cot \theta }$

2. Difference identities-
$\sin (\frac{\pi }{2}-u)=\cos u\cos (\frac{\pi }{2}-u)=\sin u\tan (\frac{\pi }{2}-u)=\cot u\text{cosec}(\frac{\pi }{2}-u)=\sec u\sec (\frac{\pi }{2}-u)=\text{cosec}u\cot (\frac{\pi }{2}-u)=\tan u$
Sine Laws-
$\frac{a}{\sin \left(A\right)}=\frac{b}{\sin \left(B\right)}=\frac{c}{\sin \left(C\right)}=2R=\frac{abc}{2\Delta }$
Area $=\Delta =\frac{1}{2ab\sin \left(C\right)}$
Cosine Laws -
$Area=\Delta =\sqrt{s(s-a)(s-b)(s-c)}=\frac{abc}{4R}$
Tangent Laws -
$\frac{a-b}{a+b}=\frac{\tan \left[\frac{1}{2}\right(A-B\left)\right]}{\tan \left[\frac{1}{2}\right(A+B\left)\right]}$

Applications of Trigonometry:
Trigonometry finds its applications in various fields like seismology, oceanography, meteorology, physical sciences, navigation, astronomy, acoustics, electronics, etc.
It is also helpful to measure the heights of the mountain, find the distance of long rivers, etc.