01
Funzioni goniometriche di angoli particolari
| Gradi | Radianti | Seno | Coseno | Tangente | Cotangente |
|---|---|---|---|---|---|
| 0° | $0$ | $0$ | $1$ | $0$ | non esiste |
| 15° | $π/12$ | $({√6-√2})/4$ | $({√6+√2})/4$ | $2-√3$ | $2+√3$ |
| 18° | $π/10$ | $({√5-1})/4$ | ${√{10+2√5}}/4$ | $√{{5-2√5}/5}$ | $√{5+2√5}$ |
| 22° 30′ | $π/8$ | ${√{2-√2}}/2$ | ${√{2+√2}}/2$ | $√2-1$ | $√2+1$ |
| 30° | $π/6$ | $1/2$ | ${√3}/2$ | ${√3}/3$ | $√3$ |
| 36° | $π/5$ | ${√{10-2√5}}/4$ | $({√5+1})/4$ | $√{5-2√5}$ | $√{{5+2√5}/5}$ |
| 45° | $π/4$ | ${√2}/2$ | ${√2}/2$ | $1$ | $1$ |
| 54° | ${3}/{10}π$ | $({√5+1})/4$ | ${√{10-2√5}}/4$ | $√{{5+2√5}/5}$ | $√{5-2√5}$ |
| 60° | $π/3$ | ${√3}/2$ | $1/2$ | $√3$ | ${√3}/3$ |
| 67° 30′ | ${3}/8π$ | ${√{2+√2}}/2$ | ${√{2-√2}}/2$ | $√2+1$ | $√2-1$ |
| 72° | ${2}/5π$ | ${√{10+2√5}}/4$ | $({√5-1})/4$ | $√{5+2√5}$ | $√{{5-2√5}/5}$ |
| 75° | ${5}/{12}π$ | $({√6+√2})/4$ | $({√6-√2})/4$ | $2+√3$ | $2-√3$ |
| 90° | $π/2$ | $1$ | $0$ | non esiste | $0$ |
| 180° | $π$ | $0$ | $-1$ | $0$ | non esiste |
| 270° | ${3}/2π$ | $-1$ | $0$ | non esiste | $0$ |
| 360° | $2π$ | $0$ | $1$ | $0$ | non esiste |
02
Relazioni fondamentali tra le funzioni goniometriche di uno stesso angolo
$$\sin^2α+\cos^2α=1$$
$$\tgα={\sinα}/{\cosα}$$
$$\cotgα={\cosα}/{\sinα}$$
$$\cosecα=1/{\sinα}$$
$$\secα=1/{\cosα}$$
03
Funzioni goniometriche di angoli associati
| Seno | Coseno | Tangente | Cotangente |
|---|---|---|---|
| $\sin(-α)=-\sinα$ | $\cos(-α)=\cosα$ | $\tg(-α)=-\tgα$ | $\cotg(-α)=-\cotgα$ |
| $\sin(2π-α)=-\sinα$ | $\cos(2π-α)=\cosα$ | $\tg(2π-α)=-\tgα$ | $\cotg(2π-α)=-\cotgα$ |
| $\sin(π-α)=\sinα$ | $\cos(π-α)=-\cosα$ | $\tg(π-α)=-\tgα$ | $\cotg(π-α)=-\cotgα$ |
| $\sin(π+α)=-\sinα$ | $\cos(π+α)=-\cosα$ | $\tg(π+α)=\tgα$ | $\cotg(π+α)=\cotgα$ |
| $\sin(π/2-α)=\cosα$ | $\cos(π/2-α)=\sinα$ | $\tg(π/2-α)=\cotgα$ | $\cotg(π/2-α)=\tgα$ |
| $\sin(π/2+α)=\cosα$ | $\cos(π/2+α)=-\sinα$ | $\tg(π/2+α)=-\cotgα$ | $\cotg(π/2+α)=-\tgα$ |
04
Formule di addizione e sottrazione
$$\sin(α+β)=\sinα\cosβ+\sinβ\cosα$$
$$\sin(α-β)=\sinα\cosβ-\sinβ\cosα$$
$$\cos(α+β)=\cosα\cosβ-\sinα\sinβ$$
$$\cos(α-β)=\cosα\cosβ+\sinα\sinβ$$
$$\tg(α+β)={\tgα+\tgβ}/{1-\tgα\tgβ}$$
con $α+β,\ α,\ β≠π/2+kπ$
$$\tg(α-β)={\tgα-\tgβ}/{1+\tgα\tgβ}$$
con $α-β,\ α,\ β≠π/2+kπ$
05
Formule di duplicazione
$$\sin2α=2\sinα\cosα$$
$$\cos2α=\cos^2α-\sin^2α=1-2\sin^2α=2\cos^2α-1$$
$$\tg2α={2\tgα}/{1-\tg^2α}$$
con $α≠π/4+k π/2 ∧ α≠π/2+kπ$
06
Formule di bisezione
$$\cos{α/2}=±√{{1+\cosα}/2}$$
$$\sin{α/2}=±√{{1-\cosα}/2}$$
$$\tg{α/2}=±√{{1-\cosα}/{1+\cosα}}$$
$(α≠π+2kπ)$
$$\tg{α/2}={\sinα}/{1+\cosα}$$
con $α≠π+2kπ$
$$\tg{α/2}={1-\cosα}/{\sinα}$$
con $α≠kπ$
07
Formule parametriche
$$\sinα={2t}/{1+t^2}$$
$$\cosα={1-t^2}/{1+t^2}$$
$(t=\tg{α/2},\ α≠π+2kπ)$
08
Formule di prostaferesi
$$\sin p+\sin q=2\sin{{p+q}/2}\cos{{p-q}/2}$$
$$\cos p+\cos q=2\cos{{p+q}/2}\cos{{p-q}/2}$$
$$\sin p-\sin q=2\cos{{p+q}/2}\sin{{p-q}/2}$$
$$\cos p-\cos q=-2\sin{{p+q}/2}\sin{{p-q}/2}$$
09
Formule di Werner
$$\sinα\cosβ=1/2[\sin(α+β)+\sin(α-β)]$$
$$\cosα\cosβ=1/2[\cos(α+β)+\cos(α-β)]$$
$$\sinα\sinβ=1/2[\cos(α-β)-\cos(α+β)]$$