TRIGONOMETRY FULL CONCEPT

 

Trigonometry Formulas

sin(−θ) = − sin θ

cos(−θ) = cos θ

tan(−θ) = − tan θ

cosec(−θ) = − cosecθ

sec(−θ) = sec θ

cot(−θ) = − cot θ

Sin( π/2- θ) = cosθ

Cos(π/2 -θ) = sin θ

tan(π/2 -θ) = cot θ

cot(π/2 -θ) = tan θ

Sec(π/2 - θ) = cosecθ

cosec(π/2 - θ) = sec θ

Sin( π/2+ θ) = cosθ Cos(π/2 +θ) = - sin θ tan(π/2 + θ) = - cot θ cot(π/2 + θ) = - tan θ Sec(π/2 + θ) = - cosecθ cosec(π/2 + θ) =  sec θ

Sin( π- θ) = Sinθ Cos(π -θ) = -Cos θ tan(π - θ) = - tan θ cot(π - θ) = - Cot θ Sec(π - θ) = - Sec θ cosec(π - θ) = Cosec θ Sin( π + θ) = - Sinθ Cos(π + θ) = - Cos θ tan(π + θ) =  tan θ cot(π + θ) =  Cot θ Sec(π + θ) = - Sec θ cosec(π + θ) = - Cosec θ

Sin( 3π/2- θ) = - cosθ Cos( 3π/2 -θ) = - sin θ tan( 3π/2 -θ) = cot θ cot( 3π/2 -θ) = tan θ Sec( 3π/2 - θ) = - cosecθ cosec( 3π/2 - θ) = - sec θ

Sin( 3π/2+ θ) = - cosθ Cos( 3π/2 +θ) =  sin θ tan( 3π/2 + θ) = - cot θ cot( 3π/2 + θ) = - tan θ Sec( 3π/2 + θ) =  cosecθ cosec( 3π/2 + θ) = - sec θ

Sin( 2π- θ) = - Sinθ Cos( 2π -θ) = Cos θ tan( 2π - θ) = - tan θ cot( 2π - θ) = - Cot θ Sec( 2π - θ) =  Sec θ cosec( 2π - θ) = - Cosec θ

Sin( 2π + θ) =  Sinθ Cos( 2π + θ) =  Cos θ tan( 2π + θ) =  tan θ cot( 2π + θ) =  Cot θ Sec( 2π + θ) =  Sec θ cosec( 2π + θ) =  Cosec θ

sin θ =perpendicular / Hypotenuse cos θ = Base / Hypotenuse tan θ = Perpendicular / Base cot θ = Base / Perpendicular sec θ = Hypotenuse / Base cosec θ = Hypotenuse /  perpendicular

Reciprocal Identities

cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ sin θ = 1/cosec θ cos θ = 1/sec θ tan θ = 1/cot θ

In Form of sin and cos:       tanθ      =  sinθ / cosθ       cotθ.    = cosθ /sinθ       secθ     = 1 / cosθ    Cosecθ  = 1 / 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

Double Angle Formulas  sin2A = 2sinA cosA            = [2tan A /(1+tan2A)]  cos2A = cos2A–sin2A              =1–2sin2A              =2cos2A–1              = [(1-tan2A)/(1+tan2A)]  tan 2A = (2 tan A)/(1-tan2A)

Thrice of Angle Formulas. sin3A = 3sinA – 4sin3A cos3A = 4cos3A – 3cosA tan3A = [3tanA–tan3A]/[1−3tan2A]

Sum and Difference Formulas hi 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)]

Product to Sum Formulas 2 Sin A Sin B =  [Cos (A-B) – Cos (A+B)] 2 Cos A Cos B =  [Cos (A-B) + Cos (A+B)] 2 Sin A Cos B =  [Sin (A+B) + Sin (A-B)] 2 Cos A Sin B =  [Sin (A+B) – Sin (A-B)] Sum to Product Formulas sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2] sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2] cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2] cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Some additional formulas for sum and product of an

• sin2A  –   sin2B  =    sin(A+B) sin(A–B)
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• cos2A –  cos2 B   =    Sin(B+A) .Sin(B–A) 
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• a.sinx + b.cosx:.   Max value =✓a²+b²
•                                 Min value = - ✓a²+b²

General Solution of Trigonometry:

Equations	Solutions
sin x = 0	x = nπ
cos x = 0	x = ( 2n + 1) π/2
tan x = 0	x = nπ
sin x = 1	x = (2nπ + π/2) = (4n+1)π/2
cos x = 1	x = 2nπ
sin x = sin θ	x = nπ + (-1)nθ, where θ ∈ [-π/2, π/2]
cos x = cos θ	x = 2nπ ± θ, where θ ∈ (0, π]
tan x = tan θ	x = nπ + θ, where θ ∈ (-π/2 , π/2]
sin ²x = sin ²θ	x = nπ ± θ
cos ²x= cos²θ	x = nπ ± θ
tan 2x = tan 2θ	x = nπ ± θ

Sinx  =  Siny.         Then   x =   nπ +(-1)