Formulario di trigonometria

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Formule fondamentali

$$\begin{array}{l} \sin^2\alpha+\cos^2\alpha=1\\ \tan x=\frac{\sin x}{\cos x}\\ \cot x=\frac{\cos x}{\sin x}\\ \sec x=\frac{1}{\cos x}\\ \csc x=\frac{1}{\sin x}\end{array}$$

Valori noti delle funzioni trigonometriche

angoli noti trigonometria

Simmetrie delle funzioni trigonometriche

$$\begin{eqnarray} \sin(-x)&=&-\sin x\\ \cos(-x)&=&\cos x\\ \tan(-x)&=&-\tan x\\ \cot(-x)&=&-\cot x\end{eqnarray}$$

Relazioni tra funzioni goniometriche elementari

$$\sin\alpha=\pm\sqrt{1-\cos^2\alpha}$$ $$\sin\alpha=\pm\frac{\tan\alpha}{\sqrt{1+\tan^2\alpha}}$$ $$\sin\alpha=\pm\frac{1}{\sqrt{1+\cot^2\alpha}}$$
$$\cos\alpha=\pm\sqrt{1-\sin^2\alpha}$$ $$\cos\alpha=\pm\frac{1}{\sqrt{1+\tan^2\alpha}}$$ $$\cos\alpha=\pm\frac{\cot\alpha}{\sqrt{1+\cot^2\alpha}}$$
$$\tan\alpha=\pm\frac{\sin\alpha}{\sqrt{1-\sin^2\alpha}}$$ $$\tan\alpha=\pm\frac{\sqrt{1-\cos^2\alpha}}{\cos\alpha}$$ $$\tan\alpha=\frac{1}{\cot\alpha}$$
$$\cot\alpha=\pm\frac{\sqrt{1-\sin^2\alpha}}{\sin\alpha}$$ $$\cot\alpha=\pm\frac{\cos\alpha}{\sqrt{1-\cos^2\alpha}}$$ $$\cot\alpha=\frac{1}{\tan\alpha}$$

Formule sugli angoli complementari e supplementari

  Angolo
Operatore $\frac{\pi}{2}-\alpha$ $\frac{\pi}{2}+\alpha$ $\pi-\alpha$ $\pi+\alpha$ $\frac{3\pi}{2}-\alpha$ $\frac{3\pi}{2}+\alpha$ $2\pi-\alpha$
$\sin$ $\cos\alpha$ $\cos\alpha$ $\sin\alpha$ $-\sin\alpha$ $-\cos\alpha$ $-\cos\alpha$ $-\sin\alpha$
$\cos$ $\sin\alpha$ $-\sin\alpha$ $-\cos\alpha$ $-\cos\alpha$ $-\sin\alpha$ $\sin\alpha$ $\cos\alpha$
$\tan$ $\cot\alpha$ $-\cot\alpha$ $-\tan\alpha$ $\tan\alpha$ $\cot\alpha$ $-\cot\alpha$ $-\tan\alpha$
$\cot$ $\tan\alpha$ $-\tan\alpha$ $-\cot\alpha$ $\cot\alpha$ $\tan\alpha$ $-\tan\alpha$ $-\cot\alpha$

Formule di addizione e sottrazione

Formule di addizione

$$\begin{array}{l} \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\\ \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta\\ \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\\ \cot(\alpha+\beta)=\frac{\cot\alpha\cot\beta-1}{\cot\beta+\cot\alpha}\end{array}$$

Formule di sottrazione

$$\begin{array}{l} \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\\ \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta\\ \tan(\alpha-\beta)=\frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\\ \cot(\alpha-\beta)=\frac{\cot\alpha\cot\beta+1}{\cot\beta-\cot\alpha}\end{array}$$

Formule di duplicazione e bisezione

Formule di duplicazione

$$\begin{array}{l} \sin2\alpha=2\sin\alpha\cos\alpha\\ \cos2\alpha=\cos^2\alpha-\sin^2\alpha\\ \tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}\\ \cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}\end{array}$$

Formule di bisezione

$$\begin{array}{l} \sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}\\ \cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}\\ \tan\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}\\ \cot\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}\end{array}$$

Formule di prostaferesi

$$\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$$ $$\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$$
$$\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}$$ $$\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}$$
$$\tan\alpha+\tan\beta=\frac{\sin(\alpha+\beta)}{\cos\alpha\cos\beta}$$ $$\tan\alpha-\tan\beta=\frac{\sin(\alpha-\beta)}{\cos\alpha\cos\beta}$$
$$\cot\alpha+\cot\beta=\frac{\sin(\beta+\alpha)}{\sin\alpha\sin\beta}$$ $$\cot\alpha-\cot\beta=\frac{\sin(\beta-\alpha)}{\sin\alpha\sin\beta}$$

Formule parametriche $(t=\tan\frac{\alpha}{2})$

$$\begin{array}{l} \sin\alpha=\frac{2t}{1+t^2}\\ \cos\alpha=\frac{1-t^2}{1+t^2}\\ \tan\alpha=\frac{2t}{1-t^2}\\ \cot\alpha=\frac{1-t^2}{2t}\end{array}$$

Formule di Werner

$$\begin{array}{l} \sin\alpha\cos\beta=\frac{1}{2}\left[\sin(\alpha+\beta)+\sin(\alpha-\beta)\right]\\ \cos\alpha\cos\beta=\frac{1}{2}\left[\cos(\alpha+\beta)+\cos(\alpha-\beta)\right]\\ \sin\alpha\sin\beta=\frac{1}{2}\left[\cos(\alpha-\beta)-\cos(\alpha+\beta)\right]\end{array}$$

Funzioni goniometriche inverse

Le funzioni goniometriche inverse sono: arcoseno, arcocoseno, arcotangente e arcocotangente.

$$\begin{array}{l} \arcsin x= \sin^{-1}x\\ \arccos x= \cos^{-1}x\\ \arctan x= \tan^{-1}x\\ arccot x= \cot^{-1}x\end{array}$$

Funzioni iperboliche

Le funzioni iperboliche sono: seno iperbolico, coseno iperbolico, tangente iperbolica, cotangente iperbolica.

Formule fondamentali

$$\begin{array}{l} \cosh^2\alpha-\sinh^2\alpha=1\\ \tanh x=\frac{\sinh x}{\cosh x}\\ \coth x=\frac{\cosh x}{\sinh x}\\ sech\ x=\frac{1}{\cosh x}\\ csch\ x=\frac{1}{\sinh x}\end{array}$$

Forma esponenziale

Per calcolare il valore delle funzioni iperboliche dobbiamo considerare la loro espressione in forma esponenziale.

$$\begin{array}{l} \sinh x=\frac{e^x-e^{-x}}{2}\\ \cosh x=\frac{e^x+e^{-x}}{2}\\ \tanh x=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\ \coth x=\frac{e^x+e^{-x}}{e^x-e^{-x}}\end{array}$$

Simmetrie delle funzioni iperboliche

$$\begin{array}{l} \sinh(-x)=-\sinh x\\ \cosh(-x)=\cosh x\\ \tanh(-x)=-\tanh x\\ \coth(-x)=-\coth x\end{array}$$

Formule di addizione e sottrazione

$$\begin{array}{l} \sinh(\alpha+\beta)=\sinh\alpha\cosh\beta+\cosh\alpha\sinh\beta\\ \cosh(\alpha+\beta)=\cosh\alpha\cosh\beta+\sinh\alpha\sinh\beta\\ \tanh(\alpha+\beta)=\frac{\tanh\alpha+\tanh\beta}{1+\tanh\alpha\tanh\beta}\\ \sinh(\alpha-\beta)=\sinh\alpha\cosh\beta-\cosh\alpha\sinh\beta\\ \cosh(\alpha-\beta)=\cosh\alpha\cosh\beta-\sinh\alpha\sinh\beta\\ \tanh(\alpha-\beta)=\frac{\tanh\alpha-\tanh\beta}{1-\tanh\alpha\tanh\beta}\end{array}$$

Formule di duplicazione e bisezione

$$\begin{array}{l} \sinh2\alpha=2\sinh\alpha\cosh\alpha\\ \cosh2\alpha=\cosh^2\alpha+\sinh^2\alpha\\ \tanh2\alpha=\frac{2\tanh\alpha}{1+\tanh^2\alpha}\\ \sinh\frac{\alpha}{2}=\pm\sqrt{\frac{\cosh\alpha-1}{2}}\\ \cosh\frac{\alpha}{2}=\sqrt{\frac{\cosh\alpha+1}{2}}\\ \tanh\frac{\alpha}{2}=\sqrt{\frac{\cosh\alpha-1}{\sinh\alpha}}\end{array}$$

Formule di prostaferesi

$$\sinh\alpha+\sinh\beta=2\sinh\frac{\alpha+\beta}{2}\cosh\frac{\alpha-\beta}{2}$$ $$\sinh\alpha-\sinh\beta=2\cosh\frac{\alpha+\beta}{2}\sinh\frac{\alpha-\beta}{2}$$
$$\cosh\alpha+\cosh\beta=2\cosh\frac{\alpha+\beta}{2}\cosh\frac{\alpha-\beta}{2}$$ $$\cosh\alpha-\cosh\beta=-2\sinh\frac{\alpha+\beta}{2}\sinh\frac{\alpha-\beta}{2}$$

Formule parametriche $(t=\tanh\frac{\alpha}{2})$

$$\begin{array}{l} \sinh\alpha=\frac{2t}{1-t^2}\\ \cosh\alpha=\frac{1+t^2}{1-t^2}\\ \tanh\alpha=\frac{2t}{1+t^2}\end{array}$$

Formule di Werner

$$\begin{array}{l} \sinh\alpha\cosh\beta=\frac{1}{2}\left[\sinh(\alpha+\beta)+\sinh(\alpha-\beta)\right]\\ \cosh\alpha\cosh\beta=\frac{1}{2}\left[\cosh(\alpha+\beta)+\cosh(\alpha-\beta)\right]\\ \sinh\alpha\sinh\beta=\frac{1}{2}\left[\cosh(\alpha-\beta)-\cosh(\alpha+\beta)\right]\end{array}$$

Funzioni iperboliche inverse

Le funzioni iperboliche inverse sono: settore seno iperbolico, settore coseno iperbolico, settore tangente iperbolica e settore cotangente iperbolica.

$$\begin{array}{l} settsinh\ x= x\log(x+\sqrt{x^2+1})& \forall x\in\mathbb R\\ settcosh\ x= x\log(x+\sqrt{x^2-1})& \mbox{con}\ x\ge 1\\ setttanh\ x= \frac{1}{2}\log\frac{1+x}{1-x}& \mbox{con}\ -1 < x < 1 \\ settcoth\ x= \frac{1}{2}\log\frac{x+1}{x-1}& \mbox{con}\ x < -1\ e\ x>1 \end{array}$$

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