I. f:R→R, f(x) = sin x II. f:R→R, f(x) = cos x1. D(f) = R2. E(f) = [-1;1]3. Zerourile: {πk ;k∈t }4. Periodicitatea: T=2π5. Semnul: + + - -

6. Paritatea: sin(-x) = -sin(x) - impara

7. Monotonia:[−π2

+2 πk ;π2+2 πk ]

[ π2

+2πk ;3 π2

+2 πk ]8. Extremele: ymin= -1 ymax= 1

1. D(f) = R2. E(f) = [-1;1]

3. Zerourile: { π2+πk ;k∈Z }

4. Periodicitatea: T=2π5. Semnul: - + - +

6. Paritatea: cos(-x) = cos(x) - para7. Monotonia:[ π +2 πk ;2π+2 πk ]; k∈Z (cadr. 3;4)

[ 2πk ; π+2πk ]; k∈Z (cadr.1,2)

8. Extremele: ymin= -1 ymax= 1

III. f:R→R, f(x) = tg x IV. f:R→R, f(x) = ctg x

1. D(f) = R\ { π2+πk | k∈Z}

2. E(f) = R3. Zerourile: {kπ ; k∈Z }4. Periodicitatea: T=π5. Semnul: - + + -

6. Paritatea:

tg(-α) = sin(−α )cos(−α) =

−sin αcosα

= - tg α -

impara

7. Monotonia:[−π2

+πk ;π2+πk ]; k∈Z

1. D(f) = R\ { πk | k∈Z}2. E(f) = R

3. Zerourile:{ π2+πk ;k∈Z }

4. Periodicitatea: T=π5. Semnul: - + + -

6. Paritatea: ctg(-α) = -ctg x; x∈D (ctg) -impara

7. Monotonia:[ πk ; π+πk ]; k∈Z

8. Extremele: Nu are extreme.

α(radiani) 0 π6

π4

π3

π2

2 π3

3 π4

5 π6

π −π6

−π4

−π3

−π2

α(grade) 0⁰ 30⁰ 45⁰ 60⁰ 90⁰ 120⁰ 135⁰ 150⁰ 180⁰ -30⁰ -45⁰ -60⁰ -90⁰

Val

oare

a

fun

cție

i sin α 0 12

√22

√32

1 √32

√22

12

0 −12

−√22

−√32

−1

cosα 1 √32

√22

12

0 −12

−√22

−√32

−1 √32

√22

12

0

tgα 0 √33

1 √3 nu exista

−√3 −1 −√33

0 −√33

−1 −√3 nu exista

ctgα nu exista

√3 1 √33

0 −√33

−¿1 −√3 nu exista

−√3 −1 −√33

0

8. Extremele: Nu are extreme.

Identitățile fundamentale ale trigonometrieisin2 α + cos2 α=1 ;α∈ R.

tgα ctgα = 1 sau tgα =1

ctgα ; sau ctgα =

1tgα

; α ≠ π2

k, k∈Z.

1+ ctg2α = 1

sin2 α ; α ≠ πk; k∈Z; 1+ tg2α =

1

cos2 α ; α ≠

π2

+ πk, k

∈Z.

Transformari elimentare ale expresiilor trigonometrice

1. sin(α + β) = sinα cosβ + sinβ cosα2. sin(α - β) = sinα cosβ - sinβ cosα3. cos(α + β) = cosα cosβ + sinα sinβ4. cos(α - β) = cosα cosβ - sinα sinβ

5. tg(α + β) = tgα+ tgβ

1−tgα tgβ

6. tg(α - β) = tgα−tgβ1+ tgα tgβ

7. ctg(α + β) = ctgα ctgβ−1ctgβ+ctgα

8. ctg(α - β) = ctgα ctgβ+1ctgβ−ctgα

9. sin2α = 2sinα cosα10. cos2α = cos2 α - sin2 α = 2cos2 α= 1= 1- sin2 α

11. tg2α = 2 tgα

1−tg2 α

12. ctg2α = ctg2 α−12 ctgα

13. sin3α = 3sinα – 4 sin3 α14. cos3α = 4cos3 α – 3cosα

15. tg3α = 3tgα−tg3 α

1−3tg2 α

16. ctg3α = ctg3−3 ctgα3 ctg2α−1