Elemente de trigonometrie. Aplicații în geometrie – teorie, formule și exemple rezolvate

Transformare grade \(\leftrightarrow\) radiani: \[\dfrac{\alpha^\circ}{180^\circ}=\dfrac{\alpha_{\text{rad}}}{\pi}\qquad\left(180^\circ=\pi\ \text{rad}\right)\]

Proprietăți: \[\sin,\cos:\mathbb{R}\to[-1,1]\Rightarrow|\sin x|\le1,\ |\cos x|\le1\]

Paritate:

\(\cos(-x)=\cos x\) (pară);\(\sin(-x)=-\sin x\), \(\operatorname{tg}(-x)=-\operatorname{tg}x\), \(\operatorname{ctg}(-x)=-\operatorname{ctg}x\) (impare)

Periodicitate:

\(\sin(x+2k\pi)=\sin x\)\(\cos(x+2k\pi)=\cos x\)\(\operatorname{tg}(x+k\pi)=\operatorname{tg}x\)\(\operatorname{ctg}(x+k\pi)=\operatorname{ctg}x\)

Reducerea la primul cadran:

\(\sin(\pi-x)=\sin x\)\(\cos(\pi-x)=-\cos x\)\(\sin\left(\dfrac{\pi}{2}-x\right)=\cos x\)

Formula fundamentală:

\(\boxed{\sin^2x+\cos^2x=1}\)\(\operatorname{tg}x=\dfrac{\sin x}{\cos x}\)\(\operatorname{ctg}x=\dfrac{\cos x}{\sin x}\)\(\operatorname{tg}x\cdot\operatorname{ctg}x=1\)

\[1+\operatorname{tg}^2x=\dfrac{1}{\cos^2x},\qquad1+\operatorname{ctg}^2x=\dfrac{1}{\sin^2x}\]

Suma și diferența:

\(\sin(a+b)=\sin a\cos b+\cos a\sin b\)\(\sin(a-b)=\sin a\cos b-\cos a\sin b\)

\(\cos(a+b)=\cos a\cos b-\sin a\sin b\)\(\cos(a-b)=\cos a\cos b+\sin a\sin b\)

\(\operatorname{tg}(a+b)=\dfrac{\operatorname{tg}a+\operatorname{tg}b}{1-\operatorname{tg}a\operatorname{tg}b}\)\(\operatorname{tg}(a-b)=\dfrac{\operatorname{tg}a-\operatorname{tg}b}{1+\operatorname{tg}a\operatorname{tg}b}\)

Unghiul dublu și unghiul pe jumătate:

\[\sin2x=2\sin x\cos x,\qquad\operatorname{tg}2x=\dfrac{2\operatorname{tg}x}{1-\operatorname{tg}^2x}\]

\[\cos2x=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x\]

\[\sin^2x=\dfrac{1-\cos2x}{2},\qquad\cos^2x=\dfrac{1+\cos2x}{2}\]

Substituția universală \(\left(t=\operatorname{tg}\dfrac{x}{2}\right)\):

\[\sin x=\dfrac{2t}{1+t^2},\qquad\cos x=\dfrac{1-t^2}{1+t^2},\qquad\operatorname{tg}x=\dfrac{2t}{1-t^2}\]

Transformarea sumelor în produse:

\(\sin a+\sin b=2\sin\dfrac{a+b}{2}\cos\dfrac{a-b}{2}\)\(\sin a-\sin b=2\sin\dfrac{a-b}{2}\cos\dfrac{a+b}{2}\)

\(\cos a+\cos b=2\cos\dfrac{a+b}{2}\cos\dfrac{a-b}{2}\)\(\cos a-\cos b=-2\sin\dfrac{a+b}{2}\sin\dfrac{a-b}{2}\)

\(\arcsin:[-1,1]\to\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\)\(\arccos:[-1,1]\to[0,\pi]\)

\(\operatorname{arctg}:\mathbb{R}\to\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\)\(\operatorname{arcctg}:\mathbb{R}\to(0,\pi)\)

\(\arcsin(-x)=-\arcsin x\)\(\arccos(-x)=\pi-\arccos x\)\(\arcsin x+\arccos x=\dfrac{\pi}{2}\)

\(\sin(\arcsin x)=x,\ x\in[-1,1]\)\(\operatorname{arctg}x+\operatorname{arcctg}x=\dfrac{\pi}{2}\)

Teorema cosinusului (în \(\triangle ABC\)): \[a^2=b^2+c^2-2bc\cos A,\qquad\cos A=\dfrac{b^2+c^2-a^2}{2bc}\]

Teorema sinusurilor: \[\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R\quad(R=\text{raza cercului circumscris})\]

Aria triunghiului oarecare:

\[\mathcal{A}=\dfrac{b\cdot h_b}{2}=\dfrac{a\cdot b\cdot\sin C}{2}=\dfrac{abc}{4R}=p\cdot r=\sqrt{p(p-a)(p-b)(p-c)},\qquad p=\dfrac{a+b+c}{2}\]

\(R=\dfrac{abc}{4\mathcal{A}}\) (cerc circumscris),\(r=\dfrac{\mathcal{A}}{p}\) (cerc înscris)

Triunghiul dreptunghic (catete \(c_1,c_2\), ipotenuză \(ip\)):

\(ip^2=c_1^2+c_2^2\) (Pitagora),\(\mathcal{A}=\dfrac{c_1c_2}{2}\),\(h=\dfrac{c_1c_2}{ip}\),\(m_{ip}=\dfrac{ip}{2}\),\(R=\dfrac{ip}{2}\)

\[\sin\alpha=\dfrac{\text{cateta opusă}}{ip},\qquad\cos\alpha=\dfrac{\text{cateta alăturată}}{ip}\]

\[\operatorname{tg}\alpha=\dfrac{\text{cateta opusă}}{\text{cateta alăturată}},\qquad\operatorname{ctg}\alpha=\dfrac{\text{cateta alăturată}}{\text{cateta opusă}}\]

Triunghiul echilateral de latură \(l\):

\(P=3l\),\(\mathcal{A}=\dfrac{l^2\sqrt{3}}{4}\),\(h=\dfrac{l\sqrt{3}}{2}\),\(R=\dfrac{l\sqrt{3}}{3}\),\(r=\dfrac{l\sqrt{3}}{6}\)