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Formule de trigonometrie

Formule de formare pentru rezolvarea problemelor matematice. identitățile de bază trigonometrice, scăderea cu formula, adunare, scădere și multiplicarea, precum și alte formule. În plus, valorile funcțiilor trigonometrice pentru cele mai comune unghiuri sunt date.

Identități de bază

\displaystyle{ \tan\alpha = \frac{\sin\alpha}{\cos\alpha} = \frac{1}{\cot\alpha} }

\displaystyle{ \cot\alpha = \frac{\cos\alpha}{\sin\alpha} = \frac{1}{\tan\alpha} }

\displaystyle{ \sin^2\alpha+\cos^2\alpha=1 }

\displaystyle{ 1 + \tan^2\alpha = \frac{1}{\cos^2\alpha} }

\displaystyle{ 1 + \cot^2\alpha = \frac{1}{\sin^2\alpha} }

\displaystyle{ \tan\alpha \cdot \cot\alpha = 1 }

Formule cu două unghiuri

\displaystyle{ \sin(2\alpha) = 2\cdot \cos\alpha \cdot \sin\alpha }

\displaystyle{ \sin(2\alpha) = \frac{2\cdot \tan\alpha}{1+\tan^2\alpha} = \frac{2\cdot \cot\alpha}{1+\cot^2\alpha} = \frac{2}{\tan\alpha+\cot\alpha} }

\displaystyle{ \cos(2\alpha) = \cos^2\alpha-\sin^2\alpha = 2\cdot \cos^2\alpha-1 = 1-2\cdot \sin^2\alpha }

\displaystyle{ \cos(2\alpha) = \frac{1-\tan^2\alpha}{1+\tan^2\alpha} = \frac{\cot^2\alpha-1}{\cot^2\alpha+1}= \frac{\cot\alpha-\tan\alpha}{\cot\alpha+\tan\alpha} }

\displaystyle{ \tan(2\alpha) = \frac{2\cdot \tan\alpha}{1-\tan^2\alpha} = \frac{2\cdot \cot\alpha}{\cot^2\alpha-1}= \frac{2}{\cot\alpha-\tan\alpha} }

\displaystyle{ \cot(2\alpha) = \frac{\cot^2\alpha-1}{2\cdot \cot\alpha} = \frac{\cot\alpha-\tan\alpha}{2} }

Formule cu trei unghiuri

\displaystyle{ \sin(3\alpha)=3\cdot \sin\alpha-4\cdot \sin^3\alpha }

\displaystyle{ \cos(3\alpha)=4\cdot \cos^3\alpha-3\cdot \cos\alpha }

\displaystyle{ \tan(3\alpha)= \frac{3\cdot \tan\alpha-\tan^3\alpha}{1-3\cdot \tan^2\alpha} }

\displaystyle{ \cot(3\alpha) = \frac{\cot^3\alpha-3\cdot \cot\alpha}{3\cdot \cot^2\alpha-1} }

Formule pentru reducerea gradului

\displaystyle{ \sin^2\alpha = \frac{1 - \cos(2\alpha)}{2} }

\displaystyle{ \cos^2\alpha = \frac{1 + \cos(2\alpha)}{2} }

\displaystyle{ \tan^2\alpha= \frac{1-\cos(2\alpha)}{1+\cos(2\alpha)} }

\displaystyle{ \cot^2\alpha= \frac{1+\cos(2\alpha)}{1-\cos(2\alpha)} }

\displaystyle{ (\sin\alpha - \cos\alpha)^2= 1-\sin(2\alpha) }

\displaystyle{ (\sin\alpha + \cos\alpha)^2= 1+\sin(2\alpha) }

Formule pentru reducerea gradului

\displaystyle{ \sin^3\alpha= \frac{3\cdot \sin\alpha-\sin(3\alpha)}{4} }

\displaystyle{ \cos^3\alpha= \frac{3\cdot \cos\alpha+\cos(3\alpha)}{4} }

\displaystyle{ \tan^3\alpha= \frac{3\cdot \sin\alpha-\sin(3\alpha)}{3\cdot \cos\alpha+\cos(3\alpha)} }

\displaystyle{ \cot^3\alpha= \frac{3\cdot \cos\alpha+\cos(3\alpha)}{3\cdot \sin\alpha-\sin(3\alpha)} }

Formule pentru reducerea gradului

\displaystyle{ \sin^4\alpha= \frac{3-4\cdot \cos(2\alpha)+\cos(4\alpha)}{8} }

\displaystyle{ \cos^4\alpha= \frac{3+4\cdot \cos(2\alpha)+\cos(4\alpha)}{8} }

\displaystyle{ \sin^5\alpha= \frac{10\cdot \sin\alpha - 5\cdot \sin(3\alpha) + \sin(5\alpha) }{16} }

\displaystyle{ \cos^5\alpha= \frac{10\cdot \cos\alpha + 5\cdot \cos(3\alpha) + \cos(5\alpha) }{16} }

Formule cu jumătate
de argument

\displaystyle{ \sin \left(\frac{\alpha}{2}\right) = \pm \sqrt{ \frac{1-\cos\alpha}{2} } }

\displaystyle{ \cos \left(\frac{\alpha}{2}\right) = \pm \sqrt{ \frac{1+\cos\alpha}{2} } }

\displaystyle{ \tan \left(\frac{\alpha}{2}\right)= \frac{1-\cos\alpha}{\sin\alpha}= \frac{\sin\alpha}{1+\cos\alpha} }

\displaystyle{ \cot \left(\frac{\alpha}{2}\right) = \frac{1+\cos\alpha}{\sin\alpha}= \frac{\sin\alpha}{1-\cos\alpha} }

Formulele de reducere
a gradului de argumentare

\displaystyle{ \sin^2 \left( \frac{\alpha}{2} \right)= \frac{1-\cos\alpha}{2} }

\displaystyle{ \cos^2 \left( \frac{\alpha}{2} \right)= \frac{1+\cos\alpha}{2} }

\displaystyle{ \tan^2 \left( \frac{\alpha}{2} \right)= \frac{1-\cos\alpha}{1+\cos\alpha} }

\displaystyle{ \cot^2 \left( \frac{\alpha}{2} \right)= \frac{1+\cos\alpha}{1-\cos\alpha} }

Formule de adiție

\displaystyle{ \sin(\alpha + \beta) = \sin\alpha \cdot \cos\beta + \cos\alpha \cdot \sin\beta }

\displaystyle{ \cos(\alpha + \beta) = \cos\alpha \cdot \cos\beta - \sin\alpha \cdot \sin\beta }

\displaystyle{ \tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha \cdot \tan\beta} }

\displaystyle{ \cot(\alpha + \beta) = \frac{\cot\alpha \cdot \cot\beta -1}{\cot\alpha + \cot\beta} }

Formulele de scădere

\displaystyle{ \sin(\alpha - \beta) = \sin\alpha \cdot \cos\beta - \cos\alpha \cdot \sin\beta }

\displaystyle{ \cos(\alpha - \beta) = \cos\alpha \cdot \cos\beta + \sin\alpha \cdot \sin\beta }

\displaystyle{ \tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \cdot \tan\beta} }

\displaystyle{ \cot(\alpha - \beta) = \frac{\cot\alpha \cdot \cot\beta +1}{\cot\alpha - \cot\beta} }

Formulele pentru conversia unei
sume în formule de produse

\displaystyle{ \sin\alpha + \sin\beta = 2\cdot \sin\left( \frac{\alpha + \beta}{2} \right) \cdot \cos\left( \frac{\alpha - \beta}{2} \right) }

\displaystyle{ \cos\alpha + \cos\beta = 2\cdot \cos\left( \frac{\alpha + \beta }{2} \right) \cdot \cos\left( \frac{\alpha - \beta }{2} \right) }

\displaystyle{ \tan\alpha + \tan\beta = \frac{\sin(\alpha + \beta) }{\cos\alpha \cdot \cos\beta} }

\displaystyle{ \cot\alpha + \cot\beta = \frac{\sin(\beta + \alpha) }{\sin\alpha \cdot \sin\beta} }

Formulele pentru conversia diferenței
în formule de produse

\displaystyle{ \sin\alpha - \sin\beta = 2\cdot \sin\left( \frac{\alpha - \beta }{2} \right) \cdot \cos\left( \frac{\alpha + \beta }{2} \right) }

\displaystyle{ \cos\alpha - \cos\beta = -2\cdot \sin\left( \frac{\alpha + \beta }{2} \right) \cdot \sin\left( \frac{\alpha - \beta }{2} \right) }

\displaystyle{ \tan\alpha - \tan\beta = \frac{\sin(\alpha - \beta) }{\cos\alpha \cdot \cos\beta} }

\displaystyle{ \cot\alpha - \cot\beta = \frac{\sin(\beta - \alpha) }{\sin\alpha \cdot \sin\beta} }

Formulele de conversie a sumelor

\displaystyle{ \sin\alpha + \cos\alpha = \sqrt{2}\cdot \sin\left( \alpha+\frac{\pi}{4} \right) }

\displaystyle{ \sin\alpha - \cos\alpha = \sqrt{2}\cdot \sin\left( \alpha-\frac{\pi}{4} \right) }

\displaystyle{ \arcsin(x) + \arccos(x) = \frac{\pi}{2} }

\displaystyle{ ar\cot(x) + arc\cot(x) = \frac{\pi}{2} }

\displaystyle{ Asin(\alpha) + Bcos(\alpha) = \sqrt{A^2+B^2}(sin(\alpha + arccos\left( \frac{A}{\sqrt{A^2+B^2}} \right))) }

\displaystyle{ Asin(\alpha) - Bcos(\alpha) = \sqrt{A^2+B^2}(sin(\alpha - arccos\left( \frac{A}{\sqrt{A^2+B^2}} \right))) }

Formulele pentru transformarea unui produs
în formule sum și diferență

\displaystyle{ \sin\alpha \cdot \sin\beta = \frac{\cos(\alpha - \beta) - \cos(\alpha + \beta)}{2} }

\displaystyle{ \sin\alpha \cdot \cos\beta = \frac{\sin(\alpha - \beta) + \sin(\alpha + \beta)}{2} }

\displaystyle{ \cos\alpha \cdot \cos\beta = \frac{\cos(\alpha - \beta) + \cos(\alpha + \beta)}{2} }

\displaystyle{ \tan\alpha \cdot \tan\beta = \frac{cos(\alpha - \beta) - \cos(\alpha + \beta)}{\cos(\alpha - \beta) + \cos(\alpha + \beta)} = \frac{\tan\alpha + \tan\beta}{\cot\alpha + \cot\beta} }

\displaystyle{ \cot\alpha \cdot \cot\beta = \frac{\cos(\alpha - \beta)+\cos(\alpha + \beta)}{\cos(\alpha - \beta)-\cos(\alpha + \beta)} = \frac{\cot\alpha + \cot\beta}{\tan\alpha + \tan\beta} }

\displaystyle{ \tan\alpha \cdot \cot\beta = \frac{\sin(\alpha - \beta)+\sin(\alpha + \beta)}{\sin(\alpha + \beta)-\sin(\alpha - \beta)} }

Formule pentru transformarea unui produs
de funcții într-o putere

\displaystyle{ \sin^2(\alpha)\cdot \cos^2(\alpha) = \frac{ 1 - \cos(4\alpha) }{8} }

\displaystyle{ \sin^3(\alpha)\cdot \cos^3(\alpha) = \frac{ 3\cdot \sin(2\alpha) - \sin(6\alpha) }{32} }

\displaystyle{ \sin^4(\alpha)\cdot \cos^4(\alpha) = \frac{ 3 - 4\cdot \cos(4\alpha) + \cos(8\alpha) }{128} }

\displaystyle{ \sin^5(\alpha)\cdot \cos^5(\alpha) = \frac{ 10\cdot \sin(2\alpha) - 5\cdot \sin(6\alpha) + \sin(10\alpha) }{512} }

Formule pentru reducerea gradului

\displaystyle{ sin^n(\alpha) = \frac{C_{\frac{n}{2}}^{n}}{2^n} + \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n}{2}-1} (-1)^{\frac{n}{2}-k} C_{k}^{n}cos((n-2k)\alpha) }

\displaystyle{ cos^n(\alpha) = \frac{C_{\frac{n}{2}}^{n}}{2^n} + \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n}{2}-1} C_{k}^{n}cos((n-2k)\alpha) }

\displaystyle{ sin^n(\alpha) = \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n-1}{2}} (-1)^{\frac{n-1}{2}-k} C_{k}^{n}sin((n-2k)\alpha) }

\displaystyle{ cos^n(\alpha) = \frac{1}{2^{n-1}} \sum_{k=0}^{\frac{n-1}{2}} C_{k}^{n}cos((n-2k)\alpha) }

Înlocuirea
trigonometrică universală

\displaystyle{ \sin\alpha = \frac{2\cdot \tan\left( \frac{\alpha}{2} \right)}{1 + \tan^2 \left( \frac{\alpha}{2} \right)} }

\displaystyle{ \cos\alpha = \frac{ 1 - \tan^2 \left( \frac{\alpha}{2} \right) }{1 + \tan^2 \left( \frac{\alpha}{2} \right)} }

\displaystyle{ \tan\alpha = \frac{2\cdot \tan\left( \frac{\alpha}{2} \right)}{1 - \tan^2 \left( \frac{\alpha}{2} \right)} }

\displaystyle{ \cot\alpha = \frac{1 - \tan^2 \left( \frac{\alpha}{2} \right)}{2\cdot \tan\left( \frac{\alpha}{2} \right)} }

Valorile funcțiilor trigonometrice

α	0	$\displaystyle{ \frac{\pi}{6} }$	$\displaystyle{ \frac{\pi}{4} }$	$\displaystyle{ \frac{\pi}{3} }$	$\displaystyle{ \frac{\pi}{2} }$	$\displaystyle{ \frac{2\pi}{3} }$	$\displaystyle{ \frac{3\pi}{4} }$	$\displaystyle{ \frac{5\pi}{6} }$	$\displaystyle{ \pi }$	$\displaystyle{ \frac{7\pi}{6} }$	$\displaystyle{ \frac{5\pi}{4} }$	$\displaystyle{ \frac{4\pi}{3} }$	$\displaystyle{ \frac{3\pi}{2} }$	$\displaystyle{ \frac{5\pi}{3} }$	$\displaystyle{ \frac{7\pi}{4} }$	$\displaystyle{ \frac{11\pi}{6} }$	$\displaystyle{ 2\pi }$
α°	0°	30°	45°	60°	90°	120°	135°	150°	180°	210°	225°	240°	270°	300°	315°	330°	360°
sin α	0	$\displaystyle{ \frac{1}{2} }$	$\displaystyle{ \frac{\sqrt{2}}{2} }$	$\displaystyle{ \frac{\sqrt{3}}{2} }$	1	$\displaystyle{ \frac{\sqrt{3}}{2} }$	$\displaystyle{ \frac{\sqrt{2}}{2} }$	$\displaystyle{ \frac{1}{2} }$	0	$\displaystyle{ -\frac{1}{2} }$	$\displaystyle{ -\frac{\sqrt{2}}{2} }$	$\displaystyle{ -\frac{\sqrt{3}}{2} }$	−1	$\displaystyle{ -\frac{\sqrt{3}}{2} }$	$\displaystyle{ -\frac{\sqrt{2}}{2} }$	$\displaystyle{ -\frac{1}{2} }$	0
cos α	1	$\displaystyle{ \frac{\sqrt{3}}{2} }$	$\displaystyle{ \frac{\sqrt{2}}{2} }$	$\displaystyle{ \frac{1}{2} }$	0	$\displaystyle{ -\frac{1}{2} }$	$\displaystyle{ -\frac{\sqrt{2}}{2} }$	$\displaystyle{ -\frac{\sqrt{3}}{2} }$	−1	$\displaystyle{ -\frac{\sqrt{3}}{2} }$	$\displaystyle{ -\frac{\sqrt{2}}{2} }$	$\displaystyle{ -\frac{1}{2} }$	0	$\displaystyle{ \frac{1}{2} }$	$\displaystyle{ \frac{\sqrt{2}}{2} }$	$\displaystyle{ \frac{\sqrt{3}}{2} }$	1
tg α	0	$\displaystyle{ \frac{1}{\sqrt{3}} }$	1	$\displaystyle{ \sqrt{3} }$	−	$\displaystyle{ - \sqrt{3} }$	−1	$\displaystyle{ -\frac{1}{\sqrt{3}} }$	0	$\displaystyle{ \frac{1}{\sqrt{3}} }$	1	$\displaystyle{ \sqrt{3} }$	−	$\displaystyle{ - \sqrt{3} }$	−1	$\displaystyle{ - \frac{1}{\sqrt{3}} }$	0
ctg α	−	$\displaystyle{ \sqrt{3} }$	1	$\displaystyle{ \frac{1}{\sqrt{3}} }$	0	$\displaystyle{ - \frac{1}{\sqrt{3}} }$	−1	$\displaystyle{ - \sqrt{3} }$	−	$\displaystyle{ \sqrt{3} }$	1	$\displaystyle{ \frac{1}{\sqrt{3}} }$	0	$\displaystyle{ - \frac{1}{\sqrt{3}} }$	−1	$\displaystyle{ - \sqrt{3} }$	−