Mr. B. R. Gupta (Govt. Teacher)
M.A., B.ED., CTET, STET
Trigonometric functions
sin α, cos α
| tan α = | sin α | , α ≠ | π | + πn, n є Z |
| cos α | 2 |
| cot α = | cos α | , α ≠ π + πn, n є Z |
| sin α |
tan α · cot α = 1
| sec α = | 1 | , α ≠ | π | + πn, n є Z |
| cos α | 2 |
| cosec α = | 1 | , α ≠ π + πn, n є Z |
| sin α |
Pythagorean identity
sin2 α + cos2 α = 1
| 1 + tan2 α = | 1 |
| cos2 α |
| 1 + cot2 α = | 1 |
| sin2 α |
Sum-Difference Formulas
sin(α + β) = sin α · cos β + cos α · sin β
sin(α – β) = sin α · cos β – cos α · sin β
cos(α + β) = cos α · cos β – sin α · sin β
cos(α – β) = cos α · cos β + sin α · sin β
| tan(α + β) = | tan α + tan β |
| 1 – tanα · tan β |
| tan(α – β) = | tan α – tan β |
| 1 + tanα · tan β |
| cot(α + β) = | cotα · cot β - 1 |
| cot β + cot α |
| cot(α - β) = | cotα · cot β + 1 |
| cot β - cot α |
Double angle formulas
sin 2α = 2 sin α · cos α
cos 2α = cos2 α - sin2 α
| tan 2α = | 2 tan α |
| 1 - tan2 α |
| cot 2α = | cot2 α - 1 |
| 2 cot α |
Triple angle formulas
sin 3α = 3 sin α - 4 sin3 α
cos 3α = 4 cos3 α - 3 cos α
| tan 3α = | 3 tan α - tan3 α |
| 1 - 3 tan2 α |
| cot 3α = | 3 cot α - cot3 α |
| 1 - 3 cot2 α |
Power-reduction formula
| sin2 α = | 1 - cos 2α |
| 2 |
| cos2 α = | 1 + cos 2α |
| 2 |
| sin3 α = | 3 sin α - sin 3α |
| 4 |
| cos3 α = | 3 cos α + cos 3α |
| 4 |
Sum (difference) to product formulas
| sin α + sin β = 2 sin | α + β | cos | α - β |
| 2 | 2 |
| sin α - sin β = 2 sin | α - β | cos | α + β |
| 2 | 2 |
| cos α + cos β = 2 cos | α + β | cos | α - β |
| 2 | 2 |
| cos α - cos β = -2 sin | α + β | sin | α - β |
| 2 | 2 |
| tan α + sin β = | sin(α + β) |
| cos α · cos β |
| tan α - sin β = | sin(α - β) |
| cos α · cos β |
| cot α + sin β = | sin(α + β) |
| sin α · sin β |
| cot α - sin β = | sin(α - β) |
| sin α · sin β |
a sin α + b cos α = r sin (α + φ),
| where r2 = a2 + b2, sin φ = | b | , tan φ = | b |
| r | a |
Product to sum (difference) formulas
| sin α · sin β = | 1 | (cos(α - β) - cos(α + β)) |
| 2 |
| sin α · cos β = | 1 | (sin(α + β) + sin(α - β)) |
| 2 |
| cos α · cos β = | 1 | (cos(α + β) + cos(α - β)) |
| 2 |
Tangent half-angle substitution
| sin α = | 2 tan (α/2) |
| 1 + tan2 (α/2) |
| cos α = | 1 - tan2 (α/2) |
| 1 + tan2 (α/2) |
| tan α = | 2 tan (α/2) |
| 1 - tan2 (α/2) |
| cot α = | 1 - tan2 (α/2) | |||||||||||||||||||||||||
2 tan (α/2)Table of cosine in radians
Table of angles cosines from 0° to 180°
Table of angles cosines from 181° to 360°
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