[Mathematics] Trigonometry Formulae For All Competitive Exams

[Advanced Mathematics]

Trigonometry Formulae For All Competitive Exams

TRIGONOMETRY

Formulae:-

• 1 right angle  = 90⁰

• 1⁰ = 60′ {1 degree = 60 minutes}

• 1′ = 60” {1 minute = 60 seconds}

• Sides ratio in functions:

L	A	L
K	K	A

P	B	P
H	H	B

L, P = Perpendicular

A, B = Base

K, H= Hypotenuse

• Sin² θ + Cos² θ = 1

• Sec² θ – Tan² θ = 1

• Cosec² θ – Cot² θ = 1

• Cos (-θ) = Cos θ

• Sec (-θ ) = Sec θ

• Tan (-θ) = – Tan θ

• Cot (-θ) = – Cot θ

• Sin (-θ) = – Sin θ

• Cosec (-θ) = – Cosec θ

Table

	00	300	450	600	900
Sin θ	0	1/2	1/√2	√3/2	1
Cos θ	1	√3/2	1/√2	1/2	0
Tan θ	0	1/√3	1	√3	∞
Cot θ	∞	√3	1	1/√3	0
Sec θ	1	2/√3	√2	2	∞
Cosec θ	∞	2	√2	2/√3	1

Quadrant	Limit	Signs of functions
I	00	All functions are positive.
II	900	Sin θ and Cosec θ are positive.
III	180	Tan θ and Cot θ are positive.
IV	270	Cos θ and Sec θ are positive.

Some Important Tricks

• m sin θ ± n cos θ {Maximum value} = √m² + n²

• m sin θ ± n sin θ {Maximum value} = √m² + n²

• m cos θ ± n cos θ {Maximum value} = √m² + n²

• tan 1⁰ . tan 2⁰ …………. tan 89⁰ = 1

• cot 1⁰ . cot 2⁰ …………. cot 89⁰ = 1

• cos 1⁰ . cos 2⁰ …………cos 90⁰ = 0

• cos 1⁰ . cos 2⁰ …………cos 90⁰  = 0

• sin 1⁰ . sin 2⁰ …………. sin 180⁰ = 0

• sin 1⁰ . sin 2⁰ …………. sin 180⁰  = 0

• Sin (A + B) =  Sin A . Cos B + Cos A. Sin B

• Sin (A – B) =  Sin A . Cos B – Cos A . Sin B

• Cos (A + B) = Cos A . Cos B – Sin A .Sin B

• Cos (A – B) = Cos A . Cos B + Sin A . Sin B

• Tan (A + B) =     tan A + tan B        

                                1 – tan A.tan B

• Tan (A – B) =    tan A – tan B      

                               1 + tan A.tan B

• Cot(A + B) =      Cot A Cot B-1       

                                  Cot B+Cot A

• Cot(A – B) =     Cot A Cot B+1        

                                Cot B – Cot A

• Sin(x+y) Sin(x-y) = Sin²x – Sin²y

• Cos(x+y) Cos(x-y) = Cos²x – Sin²y

• 2 Sin x Cos y = Sin (x+y)+ Sin(x-y)

• 2 Cos x Sin y = Sin (x+y) – Sin(x-y)

• 2 Cos x Cos y = Cos (x+y) + Cos (x-y)

• 2 Sin x Sin y = Cos (x-y) – Cos (x+y)

• Sin 2A                = 2 Sin A.Cos A

                                     =       2 tan A 

                                          1 + tan² A

• Cos 2A                  = Cos² A – Sin² A

                                     = 2 Cos² A – 1

                                     = 1 – 2Sin² A

                                     =    1 – tan² A

                                          1 + tan² A

• Tan 2A                 =        2 tan A      

                                            1 – tan² A

• Sin 3x                  = 3 Sin x- 4 Sin³x

• Cos 3x                 = 4 Cos³x- 3 Cos x

• Tan 3A                 =  3tan A –  tan³ A

                                          1 – 3 tan² A

• Sin C + Sin D = 2 Sin C + D. Cos C – D

                                             2                2

• Sin C – Sin D = 2 Sin C – D. Cos  C + D

                                              2                 2

• Cos C + Cos D = 2 Cos C + D. Cos C – D

                                                2                 2

• Cos C – Cos D = – 2 Sin C + D. Sin C – D

                                                  2               2

Sine Rule :

• Cos A = b² + c² – a²

                            2bc

• Cos B = c² + a² – b²

                           2ca

• Cos C = a²+ b² – c² 

                           2ab

• Sin θ. Sin 2θ. Sin 4θ = ¼ Sin 3θ

• Cos θ. Cos 2θ. Cos 4θ = ¼ Cos 3θ

• tan θ. tan 2θ. tan 4θ = tan 3θ

• Sin x = 2 Sin x/2. Cos x/2

• Cos x = 1 – 2 Sin² x/2

• Cos x = 2 Cos ² x/2 – 1

• Sec θ + tan θ= x, sec θ = {x²+1}/ 2x

• Sec θ – tan θ= x, sec θ = {x²+1}/ 2x

• sin θ + cos θ = x, sin θ- cos θ = √{2-x²}