# Trigonometric formulas

## 1. Identities of the trigonometric functions of same the argument:      ## 2. Values of the trigonometric functions at specific points:

α° 0 30 45 60 90 180 270 360
sin(α) 0   1 0 -1 0
cos(α) 1   0 -1 0 1
tg(α) 0 1 0 0
ctg(α) 1 0 0

## 3. Evenness or oddness:    ## 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) ±tgctg(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:      ## 8. Double-angle formulas:   ## 8. Triple-angle formulas:   ## 9. Sum and difference transformation formulas:

### Sine transformations:  ### Cosine transformations:  ### Tangent transformations:  ### Cotangent transformations:  ### b) Transformation of the sum and difference of the different trigonometric functions with different angles:  ### c) Specific formulas:    ## 10. Trigonometric functions product transformation formulas:   ## 11. Reduce trigonometric functions power formulas:    ## 11. Expression of the trigonometric functions by the tangent of the half angle:      