Trigonometric formulas

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

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:

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:

8. Double-angle formulas:

9. Triple-angle formulas:

10. Sum and difference transformation formulas:

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

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:

11. Trigonometric functions product transformation formulas:

12. Reduce trigonometric functions power formulas:

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

sinα2t1t2
tgα2t1t2
cosα1t21t2
ctgα1t22t

See also:

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