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Formule trigonometrice
- sin α =
a
c
; cos α =
b
c
; tg α =
a
b
; ctg α =
b
a
(a, b – catetele, c – ipotenuza triunghiului dreptunghic, α – unghiul, opus catetei a).
- tg α =
sin α
cos α
; ctg α =
cos α
sin α
- tg α ctg α = 1.
- sin
π
± α
= cos α; sin(π ± α) = ∓ sin α.
- cos
π
± α
= ∓ sin α; cos(π ± α) = − cos α.
- tg
π
± α
= ∓ ctg α; ctg
π
± α
= ∓ tg α.
- sec
π
± α
= ∓ cosec α; cosec
π
± α
= sec α.
- sin
2 α + cos
2 α = 1.
- 1 + tg
2 α = sec
2 α.
- 1 + ctg
2 α = cosec
2 α.
- sin(α ± β) = sin α cos β ± sin β cos α.
- cos(α ± β) = cos α cos β ∓ sin α sin β.
- tg(α ± β) =
tg α ± tg β
1 ∓ tg α tg β
- ctg(α ± β) =
ctg α ctg β ∓ 1
ctg β ± ctg α
- sin 2α = 2 sin α cos α.
- cos 2α = cos
2 α − sin
2 α.
- tg 2α =
2 tg α
1 − tg
2 α
- ctg 2α =
ctg
2 α − 1
2 ctg α
- sin 3α = 3 sin α − 4 sin
3 α.
- cos 3α = 4 cos
3 α − 3 cos α.
∣sin^
α
1 − cos α
∣cos^
α
1 + cos α
∣tg
α
1 − cos α
1 + cos α
- tg
α
sin α
1 + cos α
1 − cos α
sin α
∣ctg^
α
1 + cos α
1 − cos α
- ctg
α
sin α
1 − cos α
1 + cos α
sin α
- 1 + cos α = 2 cos
2 α
2
- 1 − cos α = 2 sin
2 α
2
- sin α ± sin β = 2 sin
α ± β
cos
α ∓ β
- cos α + cos β = 2 cos
α + β
cos
α − β
- cos α − cos β = −2 sin
α + β
sin
α − β
- tg α ± tg β =
sin(α ± β)
cos α cos β
- ctg α ± ctg β =
sin(β ± α)
sin α sin β
- sin α sin β =
[cos(α − β) − cos(α + β)].
- sin α cos β =
[sin(α + β) + sin(α − β)].
- cos α cos β =
[cos(α + β) + cos(α − β)].
- Ecuatii trigonometrice elementare:
sin x = a, |a| ≤ 1; x = (−1)
n arcsin a + πn;
cos x = a, |a| ≤ 1; x = ± arccos a + 2πn;
tg x = a, x = arctg a + πn;
ctg x = a, x = arcctg a + πn
n ∈ Z.
- arcsin x + arccos x =
π
, |x| ≤ 1.
- arctg x + arcctg x =
π
- arctg x − arctg y =
arctg
x − y
1 + xy
, daca xy > −1;
π + arctg
x − y
1 + xy
, daca x > 0 si xy < −1;
−π + arctg
x − y
1 + xy
, daca x < 0 si xy < − 1.
- 2 arcsin x =
arcsin(2x
1 − x^2 ), daca |x| ≤
π − arcsin(2x
1 − x^2 ), daca
< x ≤ 1;
−π − arcsin(2x
1 − x^2 ), daca − 1 ≤ x < −
- 2 arccos x =
[
arccos(2x
2 − 1) cand 0 ≤ x ≤ 1;
2 π − arccos(2x
2 − 1) cand − 1 ≤ x < 0.
- 2 arctg x =
arctg
2 x
1 − x^2
, daca |x| < 1;
π + arctg
2 x
1 − x^2
, daca x > 1;
−π + arctg
2 x
1 − x
2
, daca x < − 1.
arcsin x =
arcsin
1 − x
2
, daca 0 ≤ x ≤ 1;
− arcsin
1 − x^2
, daca − 1 ≤ x < 0.
arccos x = arccos
1 + x
, daca − 1 ≤ x ≤ 1.
arctg x =
arctg
1 + x^2 − 1
x
, daca x 6 = 0;
0 , daca x = 0.
