Trigonometric Ratios

Trigonometric Ratios

Trigonometrical Ratio Introduction Concept of Perpendicualr, Base, Hypotenuse Notation of Angles Relations Between Trigonometrical Ratio Trigonometrical Ratio Table

Introduction: 

"Trigonometry" is an important branch of mathematics. The word 'Trigonometry' made by two Greek words; trigonon and metron. The word trigonon means a triangle, the word metron means measure. So that we known that Trigonometry is the way were we introduced with the science of measuring an triangles. In 3rd century BC.

Concept of Perpendicular (height), Base, and Hypotenuse in a Right Angle Triangle:

For any acute angle in a right-angled triangle; Perpendicular is the side opposite to the acute angle; the side adjacent to it is called Base; Hypotenuse is the side opposite to its right angle.

Example:

Angle :

Let's Consider a BA ray. If this ray rotates through its origin B and takes the position BC, then we say that the angle ∠ABC has been generated.

The initial position AB is called the initial side of the angle and the final position BC is called the terminal side of the angle.Point B about which the ray rotated is called the vertex of the angle.

Notation of Angles :

To present an angle in a triangle we often use English alphabet, but in trigonometry we generally use  the following Greek letters which are given below:

1. 𝜽 (Theta)
   
   
   
2. 𝞪 (Alpha)
   
3. 𝞫 (Beta)
   
4. 𝜸 (Gamma)
5. 𝟇 (Phi)
6. 𝜳 (psi)
   etc.

Trigonometrical Ratios:

 The ratio between the lengths of two sides of a right-angled triangle is called a Trigonometrical Ratios.
In a right angled triangle there was three sides which give us six trigonometrical ratios.

Which are:

1. Sine → Sin,
2. Cosine → Cos,
3. Tangent → Tan,
4. Cotangent → Cot,
5. Secant → Sec, and
6. Cosecant → Cosec.

In a right-angled triangle ABC , for an acute angle 𝜽 :

 Consider an acute angle ∠ABC = 𝜽 with initial side BA. Let P be any point on the on the terminal side BC. Draw PQ  perpendicular from P on BA to get the right angled triangle AQP in which ∠ PBQ = 𝜽 .

In the right angled triangle AQP, we have

Base = BQ = x, perpendicular  = PQ = y, and

Hypotenuse = BP = r.

1. ) Sine (sin): The ratio between the lengths of perpendicular and hypotenuse are called sine (sin). 
   
2. Cosine (cos): The ratio between the lengths of base and hypotenuse are called Cosine (cos).
   
   
   
3. Tangent (tan): The ratio between the lengths of perpendicular and base are called Tangent(tan).
   
   
   
4. Cotangent (cot): The ratio between the length of base and perpendicular are called Cotangent (cot).
   
5. Secant (sec): The ratio between the length of hypotenuse and base are called Secant(sec).
   
6. Cosecant (cosec): The ratio between the length of hypotenuse and perpendicular are called Cosecant (cosec).
   

Note: sin𝛳 is a abbreviation of 'sine of angle 𝛳 ' similar to all case of other trigonometric ratios.
Tips 1): All trigonometric ratios are positive real number.
Tips 2): The above trigonometric ratios are defined for an acute angle 𝛳.

Similarly, for acute angle 𝟇  in the given right angled triangle.

1. 
   
   
2. 
   
   
   
   
3. 

Relations Between Trigonometric Ratios:

For an acute angle 𝛳. sin𝛳, cos𝛳, tan𝛳  are closely connected by an relation through trigonometry ratios. if any of them is known, the other two can be easily calculated.

From fig, We have

, , and

Now,

Reciprocal Relations:

sin 𝛳 and csc 𝛳 are reciprocal of each other.

                         

2.