Trigonometrie

1. Functii trigonometrice in triunghiul dreptunghic B

sin 퐵 = ; cos 퐵 =

c a tg 퐵 = ; ctg 퐵 =

sin 퐵 = cos 퐶 ; tg 퐵 = ctg 퐶 A C

0° 30° =휋6

45° =휋4 60° =

휋3

90° =휋2 180° = 휋 270° = 3

휋2 360° = 2휋

sin 0 12 √2

2

√32

1 0 - 1 0

cos 1 √32

√22

12 0 - 1 0 1

tg 0 √33

1 √3 − 0 − 0

ctg − √3 1 √33

0 − 0 −

2. Formula fundamental a trigonometriei

푠푖푛 푎 + 푐표푠 푎 = 1

b

3. Cercul Trigonometric

4. Functiile trigonometrice ale unghiului dublu

a) sin 2푎 = 2 ∙ sin 푎 ∙ cos 푎 b) cos 2푎 = 푐표푠 푎 − 푠푖푛 푎 = 2푐표푠 푎 − 1 = 1 − 2푠푖푛 푎

c) tg 2푎 =

d) ctg 2푎 =

I II

III IV

cos

sin α π-α

π+α 2π-α

e) 푠푖푛푎 =

f) 푐표푠푎 =

5. Formule pentru sume

a) sin(푎 ± 푏) = sin 푎 ∙ cos 푏 ± cos 푎 ∙ sin 푏 b) cos(푎 ± 푏) = cos 푎 ∙ cos 푏 ∓ sin 푎 ∙ sin 푏

c) tg(푎 ± 푏) = ± ∓ ∙

6. Transformarea sumelor in produs

a) sin 푎 + sin 푏 = 2 sin ∙ cos

b) sin 푎 − sin 푏 = 2 sin ∙ cos

c) cos 푎 + cos 푏 = 2 cos ∙ cos

d) cos 푎 − cos 푏 = −2 sin ∙ sin

7. Ecuatii trigonometrice fundamentale

a) sin 푥 = 푎 , 푎 ∈ [−1,1] ⟹ 푥 ∈ {(−1) arcsin 푎 + 푘휋|푘 ∈ ℤ} b) cos 푥 = 푎 , 푎 ∈ [−1,1] ⟹ 푥 ∈ {± arccos 푎 + 2푘휋|푘 ∈ ℤ} c) tg 푥 = 푎 , 푎 ∈ ℝ ⟹ 푥 ∈ {arctg 푎 + 푘휋|푘 ∈ ℤ} d) ctg 푥 = 푎 , 푎 ∈ ℝ ⟹ 푥 ∈ {arcctg 푎 + 푘휋|푘 ∈ ℤ}