Trigonometry

Trignometric Formulae

Relation Between Degree and Radians:

Angles in Degress03045
60
90

Angles in radians0       π / 6
π / 4π / 3π / 2

 

 1c = (180 / π)o

 

1o =  (π / 180)c

 

Trigonometric Ratios of an acute angle of a right triangle:

 

Sin θ = Length of opposite side / Length of hypotenuse side

Cos θ = Length of adjacent side / Length of hypotenuse side

Tan θ = Length of opposite side / Length of adjacent side

Sec θ = Length of hypotenuse side / Length of adjacent side

Cosec θ = Length of hypotenuse side / Length of opposite side

Cot θ = Length of adjacent side  / Length of opposite side 

 

Reciprocal Relations:

 

Sin θ = 1 / Cosec θ                       Sec θ = 1 / Cos θ

 

Cos θ = 1 / Sec θ                          Cosec θ = 1 / Sin θ

 

Tan θ = 1 / Cot θ                           Cot θ = 1 / Tan θ

  

Quotient Relations:

 

Tan θ = Sin θ / Cos θ        

 

Cot θ = Cos θ / Sin θ

 

  

Trigonometric ratios of Complementary angles:

 

Sin (90 – θ) = Cos θ                          Sec (90 – θ) = Cosec θ

 

Cos (90 – θ) = Sin θ                          Cosec (90 – θ) = Sec θ

 

Tan (90 – θ) = Cot θ                           Cot (90 – θ) = Tan θ

 

Trigonometric ratios for angle of measure:

 

  θ  0
  30
 45
  60
  90

Sin θ01/21/√2√3/21

Cos θ1√3/21/√21/20

Tan θ01/√31√3∞

Cot θ∞√311/√30

Sec θ1
2/√3√22∞

Cosec θ∞2√22/√31

  

Basic Identities

• Sin 2 x + Cos 2 x = 1

• Sec 2 x - Tan2 x = 1

• Cosec 2 x - Cot 2 x = 1

• Sin 2 x = 1 - Cos 2 x

• Cos 2 x = 1 - Sin 2 x

• Sec 2 x = 1 + Tan2x

• Cosec 2 x = 1 + Cot2 x

• Sec 2 x -1 = Tan2 x

• Cosec 2 x –1 = Cot2 x

Addition and Difference 
 

• 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)

• Sin2A = 2 SinA CosA

• Cos2A = Cos2 A - Sin2 A = 2 Cos2 A - 1 = 1 - 2 Sin2 A

• Tan2A = 2 Tan A / (1 - Tan2 A)

• Sin2 A = (1 - Cos2A)/2

• Cos2 A = (1 + Cos2A)/2

• Sin2A = 2TanA/ (1+Tan2A)

• Cos2A = (1-Tan2A)/ (1 + Tan2A)

• Tan2A = 2TanA/ (1-Tan2A)

• Sin A = 2 SinA/2 CosA/2 = 2TanA/2 / (1+Tan2A/2)

• Cos A = Cos2 A/2 - Sin2 A/2 = 2 Cos2 A/2 - 1 = 1 - 2 Sin2 A/2 = (1-Tan2A/2)/ (1 + Tan2A/2)

• TanA = 2TanA/2 / (1-Tan2A/2)

• Sin 3A = 3 Sin A - 4 Sin3A

• Cos 3A = 4 Cos3A- 3 Cos A

• Tan 3A = (3Tan A – tan3A) / (1-3Tan2A)

• Cos2A/2 = 1 + Cos A/2

• Sin2A/2 = 1 - Cos A/2

• Cos3A = (3Cos A + Cos 3A)/4

• Sin3A = (3Sin A + Sin 3A)/4

• Sin 18 = (√5 – 1)/ 4

• Cos 36 = (√5 + 1)/ 4

• Sin(A+B) Sin(A-B) = Sin2 A - Sin2 B

• Cos(A+B) Cos(A-B) = Cos2 A - Sin2 B

• 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 )

• Cos C - Cos D = -2 Sin( (C - D)/2 ) Sin( (C + D)/2 )

• Sin C + Sin D = 2 Sin( (C + D)/2 ) Cos( (C - D)/2 )

• Cos C + Cos D = 2 Cos( (C - D)/2 ) Cos( (C + D)/2 )

   
Hyperbolic Definitions

sinh(x) = ( e x - e -x )/2

cosech(x) = 1/sinh(x) = 2/( e x - e -x )

cosh(x) = ( e x + e -x )/2

sech(x) = 1/cosh(x) = 2/( e x + e -x )

tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )

coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )

cosh 2(x) - sinh 2(x) = 1

tanh 2(x) + sech 2(x) = 1

coth 2(x) - cosech 2(x) = 1

 

Inverse Hyperbolic Definitions

Sinh-1(z) = log( z + (√z2 + 1) )
Cosh-1(z) = log( z + (√z2 - 1) )

Tanh-1(z) = 1/2 log( (1+z)/(1-z) )

 

 

Relations to Trigonometric Functions

sinh(z) = -i sin(iz)

cosech(z) = i cosec(iz)

cosh(z) = cos(iz)

sech(z) = sec(iz)

tanh(z) = -i tan(iz)

coth(z) = i cot(iz)

sin(-x) = -sin(x)

cosec(-x) = -cosec(x)

cos(-x) = cos(x)

sec(-x) = sec(x)

tan(-x) = -tan(x)

cot(-x) = -cot(x)