Relation Between Degree and Radians:

Angles in Degress	0	30	45	60	90
Angles in radians	0	π / 6	π / 4	π / 3	π / 2

 1^c = (180 / π)^o

 

1^o =  (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 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

 

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 )