Trigonometry Formulas

Trigonometry Formulas:

Trigonometric Ratios Formulas:

Circular Functions or Trigonometric Ratios
sin θ = Perpendicular/Hypotenuse = MP/OP = y/r
cos θ = Adjacent Side or Base/Hypotenuse = OM/OP = x/r
tan θ = Perpendicular/Adjacent Side or Base = MP/OM = y/x
cosec θ = Hypotenuse/Perpendicular = OP/MP = r/y
sec θ = Hypotenuse/Adjacent Side or Base = OP/OM = r/x
cot θ = Adjacent Side or Base/Perpendicular = OM/MP =x/y

Relation Between Trigonometric Ratios:

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

Trigonometric Identities:

sin2 θ + cos2 θ = 1
1 + tan2 θ = sec2 θ
1 + cot2 θ = cosec2 θ

Trigonometry Table:

Radian Measure = (π/180) x Degree Measure
Degree Measure = (180/π) x Radian Measure
Angles (In Radians)0π/6π/4π/3π/2π3π/2
Angles (In Degrees)30°45°60°90°180°270°360°
sin01/21/√2√3/210-10
cos1√3/21/√21/20-101
tan01/√31√3
00
cot
√311/√300
sec12/√3√22
-11
cosec
2√22/√31-1

Trigonometric Functions:

Circular Functions Quadrants
IIIIIIIV
sin θ+ve+ve-ve-ve
cos θ+ve-ve-ve+ve
tan θ+ve-ve+ve-ve
cosec θ+ve+ve-ve-ve
sec θ+ve-ve-ve+ve
cot θ+ve-ve+ve-ve
sin (π/2 – θ) = cos θsin (π/2 + θ) = cos θ
cos (π/2 – θ) = sin θcos (π/2 + θ) = – sin θ
tan (π/2 – θ) = cot θtan (π/2 + θ) = – cot θ
cot (π/2 – θ) = tan θcot (π/2 + θ) = – tan θ
sec (π/2 – θ) = cosec θsec (π/2 + θ) = – cosec θ
cosec (π/2 – θ) = sec θcosec (π/2 + θ) = sec θ
sin (π – θ) = sin θsin (π + θ) = – sin θ
cos (π – θ) = – cos θcos (π + θ) = – cos θ
tan (π – θ) = – tan θtan (π + θ) = tan θ
cot (π – θ) = – cot θcot (π + θ) = cot θ
sec (π – θ) = – sec θsec (π + θ) = – sec θ
cosec (π – θ) = cosec θcosec (π + θ) = – cosec θ
sin (-x) = – sin xcos (-x) = cos x

Addition and Subtraction Formulas in Trigonometry:

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 (A + B). sin (A- B) = sin2 A – sin2 B
cos (A + B). cos (A- B) = cos2 A – sin2 B = cos2 B – sin2 A
sin (A + B + C) = cos A cos B cos C (tan A + tan B + tan C – tan A tan B tan C)
cos (A + B + C) = cos A cos B cos C (1- tan A tan B – tan B tan C – tan C tan A)
tan (A + B + C) = (tan A + tan B + tan C – tan A tan B tan C)/(1- tan A tan B – tan B tan C – tan C tan A)

Sum and Difference of Sines or Cosines:

2 sin A cos B = sin (A + B) + sin (A – B)
2 cos A sin B = sin (A + B) – sin (A – B)
2 cos A cos B = cos (A + B) + cos (A – B)
2 sin A sin B = cos (A- B) – cos (A + B)
sin C + sin D = 2 sin (C + D)/2 . cos (C – D)/2
sin C – sin D = 2 cos (C + D)/2 . sin (C – D)/2
cos C + cos D = 2 cos (C + D)/2 . cos (C – D)/2
cos C – cos D = 2 sin (C + D)/2 . sin (D – C)/2

Trigonometrical Ratios of Multiple Angles:

sin 2A = 2 sin A cos A = 2 tan A/(1 + tan2 A)
cos 2A = cos2 A – sin2 A = 2 cos2 A – 1 = 1 – 2 sin2 A = (1 – tan2 A)/(1 + tan2 A)
tan 2A = 2 tan A/(1 – tan2 A)
cot 2A = (cot2 A – 1)/2cot A
sin 3A = 3 sin A – 4 sin3 A
cos 3A = 4 cos3 A – 3 cos A
tan 3A = (3 tan A – tan3 A)/(1 – 3 tan2 A)

Sine Formula:

In any triangle ABC, a/sin A = b/sin B = c/sin C 

Cosine Formula:

cos A = (b2 + c2 – a2)/2bc
cos B = (c2 + a2 – b2)/2ca
cos C = (a2 + b2 – c2)/2ab

Projection Formula:

a = b cos C + c cos B
b = c cos A + a cos C
c = a cos B + b cos A

Addition and Subtraction Formulas in Trigonometry
Converting Sum or Difference into Product in Trigonometry
Trigonometrical Ratios of Multiple Angles
Trigonometric Ratios of Submultiple Angles
Trigonometric Identities Connecting the Angles of a Triangle
Graphs of Simple Trigonometric Functions
Projection Formulae and Area of a Triangle
List of trigonometric identities– Wikipedia

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