Trigonometric formulas

1. Identities of the trigonometric functions of the same argument:

tgαsinαcosα
sin2αcos2α1
1tg2α1cos2α
ctgαcosαsinα
tgαctgα1
1ctg2α1sin2α

2. Values of the trigonometric functions at specific points:

α° 0 30 45 60 90 180 270 360
sin(α) 0 12 22 32 1 0 -1 0
cos(α) 1 32 22 12 0 -1 0 1
tg(α) 0 13 1 3 0 0
ctg(α) 3 1 13 0 0

3. Evenness or oddness:

sinαsinα
cosαcosα
tgαtgα
ctgαctgα

4. The signs at quarters:

Trig. function Quater
I II III IV
sin(α) + +
cos(α) + +
tg(α) + +
ctg(α) + +

5. Reduction formulas:

β = 90 ± α 180 ± α 270 ± α 360 ± α
sin(β) cos(α) ∓sin(α) cos(α) ±sin(α)
cos(β) ∓sin(α) cos(α) ±sin(α) cos(α)
tg(β) ∓ctg(α) ±tg(α) ∓ctg(α) ±tg(α)
ctg(β) ∓tg(α) ±ctg(α) ∓tg(α) ∓ctg(α)

6. Solution of the simplest trigonometric equations:

sin(x) = a ⇔ x = (-1)n arcsin(a) + n π, n = 0, 1 ... ∈ Z

cos(x) = a ⇔ x = ± arccos(a) + 2 n π, n = 0, 1 ... ∈ Z

tg(x) = a ⇔ x = arctg(a) + n π, n = 0, 1 ... ∈ Z

7. Sum and subtraction formulas:

sinαβsinαcosβcosαsinβ
sinαβsinαcosβcosαsinβ
cosαβcosαcosβsinαsinβ
cosαβcosαcosβsinαsinβ
tgαβtgαtgβ1tgαtgβ
tgαβtgαtgβ1tgαtgβ

8. Double-angle formulas:

sin2α2sinαcosα
cos2αcos2αsin2α
tg2α2tgα1tg2α

9. Triple-angle formulas:

sin3α3sinα4sin3α
cos3α4cos3α3cosα
tg3α3tgαtg3α13tg2α

10. Sum and difference transformation formulas:

a) Transformation of the sum and difference of the same trigonometric functions with different angles:

Sine transformations:

sinαsinβ2sinαβ2cosαβ2
sinαsinβ2sinαβ2cosαβ2

Cosine transformations:

cosαcosβ2cosαβ2cosαβ2
cosαcosβ2sinαβ2sinαβ2

Tangent transformations:

tgαtgβsinαβcosαcosβ
tgαtgβsinαβcosαcosβ

Cotangent transformations:

ctgαctgβsinβαsinαsinβ
ctgαctgβsinβαsinαsinβ

b) Transformation of the sum and difference of the different trigonometric functions with different angles:

sinαcosβ2sinαβ2π4cosαβ2π4
sinαcosβ2sinαβ2π4cosαβ2π4

c) Specific formulas:

sinαcosα2sinαπ4
AsinαBcosαA2B2sinαarctgBA
sinαsin2α...sinnαsin12n1αsinnα2sinα2
cosαcos2α...cosnαcos12n1αsinnα2sinα2

11. Trigonometric functions product transformation formulas:

sinαsinβcosαβcosαβ2
sinαcosβsinαβsinαβ2
cosαcosβcosαβcosαβ2

12. Reduce trigonometric functions power formulas:

sin2α1cos2α2
cos2α1cos2α2
sin3α3sinαsin3α4
cos3α3cosαcos3α4

13. Expression of the trigonometric functions by the tangent of the half angle:

ttgα2sinα1cosα1cosαsinα1cosα1cosα

sinα2t1t2
tgα2t1t2
cosα1t21t2
ctgα1t22t

Other useful links:

Partial derivative step by step sample
Formulas of arithmetical and geometrical progressions
Logarithms identities

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