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 |
|
|
|
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) |
∓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: