Formule trigonometrice

● Tabel de valori:

t 0

sin t 0 1 0 -1 0

cos t 1 0 -1 0 1

tg t 0 1 - 0 - 0

ctg t - 1 0 - 0 -

● Formula fundamentală a trigonometrieicos2t+sin2 t=1 ,∀ t∈R

sin (t+2kπ )=sin t ,∀ t∈ R ,∀ k∈Z

cos (t+2kπ )=cos t ,∀ t∈R ,∀ k∈Z

t g x= sin xcos x

, x∈ R ¿{(2k+1 ) π2|k∈Z¿}

c t g x= cos xsin x

, x∈R ¿{kπ|k∈Z ¿}

● Paritatea functiei cos si imparitatea functiilor sin, tg, ctgcos (−t )=cos t , t∈R sin (−t )=−sin t ,t∈R

tg (−t )=−tg t ,t ≠ (2k+1 ) π2

ctg (−t )=−ctg t , t ≠ kπ

● Trecerea la cofunctie

sin( π2 −x)=cos x

cos( π2−x )=sin x

cos (a−b )=cos a ∙cos b+sina ∙ sinbcos (a+b)=cos a∙cos b−sin a∙ sinbsin(a+b)=sina ∙cos b+sinb ∙cos asin(a−b)=sina ∙cosb−sinb ∙cosa

● Sinusul si cosinusul unghiului dublusin 2 x=2 sin x cos xcos2 x=cos2 x−sin2 xcos2 x=1−2sin2 xcos2 x=2 cos2 x−1

● Formule universale

sin x=2 tg

x2

1+tg2 x2

cos x=1−tg2 x

2

1+tg2 x2

tg x=2tg

x2

1−tg2 x2

ctg x=1−tg2 x

2

2 tgx2

sin x=± tg x

√1+tg2 x

cos x=± 1

√1+ tg2 x

tg x= sin 2 x1+cos 2x

tg x=1−cos2 xsin 2 x

● Transformarea produselor în sume

sin x ∙cos y=sin ( x+ y )+sin (x− y )

2

cos x ∙cos y=cos ( x+ y )+cos (x− y )

2

sin x ∙ sin y=cos ( x− y )−cos (x+ y)

2

● Transformarea sumelor în produse

sina+sinb=2 sina+b

2∙cos

a−b2

cos a+cosb=2 cosa+b

2∙cos

a−b2

cos a−cos b=−2sina+b

2∙ sin

a−b2

sina−sinb=2 sina−b

2∙cos

a+b2

●Relatii intre sin si cos

cos ( π2 +x)=−sin x sin(π−x)= sin x

sin(2π−x)= −¿sin x cos(π−x )=−¿ cos x

cos kπ=(−1)k

sin kπ=0

sin(π+x)= −¿sin x sin( π2 +x )=cos x

cos(2π−x)= cos x cos(π+x)= −¿cos x