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:

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)