Trigonometry Basic Concept

Trigonometry is an another important chapter in Mathematics for all category students or competitors, who are struggling for cracking SSC, Bank Or Railways and etc for their dream job due lack of proper guidance. First of all I would like to clear you the base concept of this chapter.

Trigonometry :- Trigonometry is a branch of Mathematics which deals in triangles and the relationship between their sides and angles and these relationships can be made using the trigonometric functions and ratios.

Angle:- When two straight non-parallel lines in a plain intersect each other, the inclination formed between them is known as angle between those two lines.                

                                           

Triangle :- Triangle is a figure bounded by three non- parallel straight lines intersecting each other, having three vertices, angles with the condition that “Sum of all interior angles is equal to 180⁰.

         i.e.                         angle A +angle B + angle C = 180⁰

Triangles are classified under two sections :-

·         Classification based on sides.

·         Classification based on angles.

Classification Based on Sides :- Triangles are of three types under this category of classification :-

1.      Scalene Triangle (Triangle with all sides of different length.)

2.      Isosceles Triangles (Triangle with any two sides of same length)

3.      Equilateral Triangles (Triangle with all sides of same length and angles of same measure 60⁰)

Classification Based on Sides :- Triangles are of three types under this category of classification :-

1.      Acute- angled Triangle (Triangle with all angles of different measure less than 90⁰)

2.      Right- angled Triangle (Triangle with any one angle of measure 90⁰.)

3.      Obtuse-angled Triangle (Triangle with any one angle of measure more than 90⁰)

Right- Angled Triangle :-Right-angled triangle is a figure bounded by three non- parallel straight lines intersecting each other such that any two lines intersecting perpendicularly (means angle formed between them is 90⁰) and having three vertices, angles with the condition that “Sum of all interior angles is equal to 180⁰.

                                   

In the above figure

                                   Side AB is known as Perpendicular denoted as “P”

                                   Side BC is known as Hypotenuse denoted as “H”

                                   Side BC is known as Base denoted as “B”

Trigonometric Ratios: The ratio between two sides of right-angled triangle are known as trigonometric ratios. The base ratios are

                                                       Sinθ = P/H

                                                      Cosθ = B/H

                                                      Tanθ = Sinθ/Cosθ = (P/H) × (H/B) = P/B

The above three ratios have one brother each which are related inversely with them. Let’s see this relationship :-

                                  Cosecθ = 1/Sinθ = H/P

                                  Secθ = 1/Cosθ = H/B

                                  Cotθ = 1/Tanθ = Cosθ/ Sinθ = (B/H) × (P/H) = B/P.

  Hence we can say that

                    Sinθ and Cosecθ are brothers which are related to each-other inversely.

                      Cosθ and Secθ are brothers which are related to each-other inversely.

                    Tanθ and Cotθ are brothers which are related to each-other inversely.

Sinθ × Cosecθ = 1 Cosθ × Secθ = 1 Tanθ × Cotθ = 1

           This implies that     

  Table for the value of Trigonometric ratios at various angles

Angle → Ratio ▼	θ =  0	θ = 300	θ = 450	θ = 600	θ = 900
Sinθ	0	½	1/√2	√3/2	1
Cosθ	1	√3/2	1/√2	½	0
Tanθ	0	1/√3	1	√3	∞
Cosecθ	∞	2	√2	2/√3	1
Secθ	1	2/√3	√2	2	∞
Cotθ	∞	√3	1	1/√3	0

Quadrant rule

Summary of Trigonometric ratios  with respect to Quadrant rule

The Quadrant rule is also known as A-S-T-C Rule

     Where A stands for all ratios means (Sinθ, Cosθ, Tanθ, Cosecθ, Secθ,  and Cotθ)

                  S stands for ratios (Sinθ and its inversely related brother Cosecθ only)

                  T stands for ratios (Tanθ and its inversely related brother Cotθ only)

                  C stands for ratios (Cosθ and its inversely related brother Secθ only)

Means :-

All ratios are positive in 1^st Quadrant

Only Sinθ and Cosecθ values are positive in 2^nd Quadrant while all others are negative in this Quadrant as this Quadrant belongs to these two ratios only.

Only Tanθ and Cotθ values are positive in 3^rd  Quadrant while all others are negative in this Quadrant as this Quadrant belongs to these two ratios only.

Only Cosθ and Secθ values are positive in 2^nd Quadrant while all others are negative in this Quadrant as this Quadrant belongs to these two ratios only.

Table for Quadrant Rule

Angles	(90 - θ)	(90 + θ)	(180 - θ)	(180 + θ)	(360 - θ) or (-θ)
Sinθ	Cosθ	Cosθ	Sinθ	-Sinθ	-Sinθ
Cosθ	Sinθ	- Sinθ	-Cosθ	-Cosθ	Cosθ
Tanθ	Cotθ	-Cotθ	-Tanθ	Tanθ	-Tanθ
Cosecθ	Secθ	Secθ	Cosecθ	-Cosecθ	-Cosecθ
Secθ	Cosecθ	-Cosecθ	-Secθ	-Secθ	Secθ
Cotθ	Tanθ	-Tanθ	-Cotθ	Cotθ	-Cotθ

Trigonometric Functions of Acute Angles

Sinθ = P/H

Cosθ = B/H

Tanθ = P/B

Cosecθ = H/P

Secθ = H/B

Cotθ = B/P.

Relations between Trigonometric Functions

Sinθ = 1/Cosecθ

Cosθ = 1/Secθ

Tanθ = 1/Cotθ

Cosecθ = 1/Sinθ

Secθ = 1/Cosθ

Cotθ = 1/Tanθ

Tanθ = Sinθ/Cosθ

Cotθ = Cosθ/ Sinθ

Pythagorean Identities

Sin²θ + Cos²θ = 1           →  Sin²θ = 1 - Cos²θ               →  Cos²θ = 1 -  Sin²θ    

Sec²θ - Tan²θ = 1           →  Sec²θ = 1 + Tan²θ               →  Sec²θ -  1 = Tan²θ    

Cosec²θ - Cot²θ = 1      →  Cosec²θ = 1 + Cot²θ           →  Cosec²θ -  1 = Cot²θ  

Addition Formulas

Sin(A + B) = SinA CosB + CosA sinB

Sin(A - B) = SinA CosB  - CosA SinB

Cos(A + B) = CosA CosB - SinA SinB

Cos(A - B) = CosA CosB + SinA SinB

Tan(A + B) = [ TanA + TanB ] / [ 1 - TanA TanB]

Tan(A + B) = [ TanA - TanB ] / [ 1 + TanA TanB]

Cot(A + B) = [ CotA CotB - 1 ] / [ CotA + CotB]

Cot(A + B) = [ CotA CotB - 1 ] / [ CotA + CotB] 

Product to Sum/Difference Formulas

SinA CosB = (1/2) [ Sin (A + B) + Sin (A - B)] 

CosA SinB = (1/2) [ Sin (A + B) - Sin (A - B)] 

CosA CosB = (1/2) [ Cos (A + B) + Cos (A - B)] 

SinA SinB = (1/2) [ Cos (A - B) - Cos (A + B)] 

Consider A+B = C & A – B = D in above Formulae

Sum to Product Formulas

SinC + SinD = 2Sin[ (C + D) / 2 ] Cos[ (C - D) / 2 ]

CosC + CosD = 2Cos[ (C + D) / 2 ] Cos[ (C - D) / 2 ] 

Difference to Product Formulas

SinC - SinD = 2Cos[ (C + D) / 2 ] Sin[ (C - D) / 2 ]

CosC - CosD = 2Sin[ (C + D) / 2 ] Sin[ (D - C) / 2 ] 

Difference of Squares Formulas

Sin ²A - Sin ²B = Sin(A + B) Sin(A - B)

Cos ²B - Cos ²A = Sin(A + B) Sin(A - B) 

Cos ²A - Sin ²B = Cos(A + B) Cos(A - B)

Cos ²B - Sin ²A = Cos(A + B) Cos(A - B)

More Concept and Related Questions will be continued soon............ till then prepare this.