Tablice z wartościami funkcji trygonometrycznych dla kątów ostrych znajdują się
pod tym linkiem.
Jedynka trygonometryczne
\[\sin^2{\alpha }+\cos^2{\alpha }=1\]
Wzory na tangens i cotangens
\[\begin{split} &\text{tg}{\alpha }=\frac{\sin{\alpha }}{\cos{\alpha}}\\[12pt] &\text{ctg}{\alpha}=\frac{\cos{\alpha}}{\sin{\alpha}}\\[12pt] &\text{tg}{\alpha}\cdot \text{ctg}{\alpha=1} \end{split}\]
Funkcje trygonometryczne podwojonego kąta
\[\begin{split} &\sin{2\alpha }=2\sin{\alpha }\cos{\alpha }=\frac{2\ \text{tg}{\alpha }}{1 +\text{tg}^2{\alpha }}\\[12pt] &\cos{2\alpha }=\cos{^2\alpha }-\sin{^2\alpha}=2\cos^2\alpha-1\\[12pt] &\text{tg}{2\alpha }=\frac{2\ \text{tg}{\alpha }}{1-\text{tg}^2{\alpha }}=\frac{2}{\text{ctg}{\alpha }-\text{tg}{\alpha }}\\[12pt] &\text{ctg}{2\alpha }=\frac{\text{ctg}^2{\alpha }-1}{2\ \text{ctg}{\alpha }}=\frac{\text{ctg}{\alpha }-\text{tg}{\alpha }}{2}\\[12pt] \end{split}\]
Funkcje trygonometryczne potrojonego kąta
\[\begin{split} &\sin{3\alpha }=-4\sin^3{\alpha }+3\sin{\alpha }\\[12pt] &\cos{3\alpha }=4 \cos^3{\alpha }-3\cos{\alpha }\\[12pt] &\text{tg}{3\alpha }=\frac{3\ \text{tg}{\alpha }-\text{tg}^3{\alpha }}{1-3\ \text{tg}^2{\alpha }}\\[12pt] &\text{ctg}{3\alpha }=\frac{\text{ctg}^3{\alpha }-3\ \text{ctg}{\alpha }}{3\ \text{ctg}^2{\alpha }-1} \end{split}\]
Funkcje trygonometryczne sumy i różnicy kątów
\[ \sin{\left ( \alpha +\beta \right )}=\sin{\alpha }\cos{\beta }+\sin{\beta }\cos{\alpha }\\[12pt] \sin{\left ( \alpha -\beta \right )}=\sin{\alpha }\cos{\beta }-\sin{\beta }\cos{\alpha }\\[12pt] \cos{\left ( \alpha +\beta \right )}=\cos{\alpha }\cos{\beta }-\sin{\alpha }\sin{\beta }\\[12pt] \cos{\left ( \alpha -\beta \right )}=\cos{\alpha }\cos{\beta }+\sin{\alpha }\sin{\beta }\\[12pt] \text{tg}{\left ( \alpha +\beta \right )}=\frac{\text{tg}{\alpha }+\text{tg}{\beta }}{1-\text{tg}{\alpha }\ \text{tg}{\beta }}\\[12pt] \text{tg}{\left ( \alpha -\beta \right )}=\frac{\text{tg}{\alpha }-\text{tg}{\beta }}{1+\text{tg}{\alpha }\ \text{tg}{\beta }}\\[12pt] \text{ctg}{\left ( \alpha +\beta \right )}=\frac{\text{ctg}{\alpha }\ \text{ctg}{\beta }-1}{\text{ctg}{\beta }+\text{ctg}{\alpha }}\\[12pt] \text{ctg}{\left ( \alpha -\beta \right )}=\frac{\text{ctg}{\alpha }\ \text{ctg}{\beta }+1}{\text{ctg}{\beta }-\text{ctg}{\alpha }}\\[12pt] \] Wzory redukcyjne
| \[\begin{split}&\sin{\left ( 90^\circ +\alpha \right )}=\cos{\alpha }\\[12pt] &\cos{\left ( 90^\circ +\alpha \right )}=-\sin{\alpha }\\[12pt] &\text{tg}{\left ( 90^\circ +\alpha \right )}=-\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 90^\circ +\alpha \right )}=-\text{tg}{\alpha } \end{split}\] | | \[\begin{split} &\sin{\left ( 90^\circ -\alpha \right )}=\cos{\alpha }\\[12pt] &\cos{\left ( 90^\circ -\alpha \right )}=\sin{\alpha }\\[12pt] &\text{tg}{\left ( 90^\circ -\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 90^\circ -\alpha \right )}=\text{tg}{\alpha } \end{split}\] |
| \[\begin{split} &\sin{\left ( 180^\circ +\alpha \right )}=-\sin{\alpha }\\[12pt] &\cos{\left ( 180^\circ +\alpha \right )}=-\cos{\alpha }\\[12pt] &\text{tg}{\left ( 180^\circ +\alpha \right )}=\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 180^\circ +\alpha \right )}=\text{ctg}{\alpha } \end{split}\] | | \[\begin{split} &\sin{\left ( 180^\circ -\alpha \right )}=\sin{\alpha }\\[12pt] &\cos{\left ( 180^\circ -\alpha \right )}=-\cos{\alpha }\\[12pt] &\text{tg}{\left ( 180^\circ -\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 180^\circ -\alpha \right )}=-\text{ctg}{\alpha } \end{split}\] |
| \[\begin{split} &\sin{\left ( 270^\circ +\alpha \right )}=-\cos{\alpha }\\[12pt] &\cos{\left ( 270^\circ +\alpha \right )}=\sin{\alpha }\\[12pt] &\text{tg}{\left ( 270^\circ +\alpha \right )}=-\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 270^\circ +\alpha \right )}=-\text{tg}{\alpha } \end{split}\] | | \[\begin{split} &\sin{\left ( 270^\circ -\alpha \right )}=-\cos{\alpha }\\[12pt] &\cos{\left ( 270^\circ -\alpha \right )}=-\sin{\alpha }\\[12pt] &\text{tg}{\left ( 270^\circ -\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 270^\circ -\alpha \right )}=\text{tg}{\alpha } \end{split}\] |
| \[\begin{split} &\sin{\left ( 360^\circ +\alpha \right )}=\sin{\alpha }\\[12pt] &\cos{\left ( 360^\circ +\alpha \right )}=\cos{\alpha }\\[12pt] &\text{tg}{\left ( 360^\circ +\alpha \right )}=\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 360^\circ +\alpha \right )}=\text{ctg}{\alpha } \end{split}\] | | \[\begin{split} &\sin{\left ( 360^\circ -\alpha \right )}=-\sin{\alpha }\\[12pt] &\cos{\left ( 360^\circ -\alpha \right )}=\cos{\alpha }\\[12pt] &\text{tg}{\left ( 360^\circ -\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 360^\circ -\alpha \right )}=-\text{ctg}{\alpha } \end{split}\] |
\[\begin{split}&\sin{\left ( 90^\circ +\alpha \right )}=\cos{\alpha }\\[12pt] &\cos{\left ( 90^\circ +\alpha \right )}=-\sin{\alpha }\\[12pt] &\text{tg}{\left ( 90^\circ +\alpha \right )}=-\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 90^\circ +\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\sin{\left ( 90^\circ -\alpha \right )}=\cos{\alpha }\\[12pt] &\cos{\left ( 90^\circ -\alpha \right )}=\sin{\alpha }\\[12pt] &\text{tg}{\left ( 90^\circ -\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 90^\circ -\alpha \right )}=\text{tg}{\alpha }\\[22pt] &\sin{\left ( 180^\circ +\alpha \right )}=-\sin{\alpha }\\[12pt] &\cos{\left ( 180^\circ +\alpha \right )}=-\cos{\alpha }\\[12pt] &\text{tg}{\left ( 180^\circ +\alpha \right )}=\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 180^\circ +\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\sin{\left ( 180^\circ -\alpha \right )}=\sin{\alpha }\\[12pt] &\cos{\left ( 180^\circ -\alpha \right )}=-\cos{\alpha }\\[12pt] &\text{tg}{\left ( 180^\circ -\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 180^\circ -\alpha \right )}=-\text{ctg}{\alpha }\\[22pt] &\sin{\left ( 270^\circ +\alpha \right )}=-\cos{\alpha }\\[12pt] &\cos{\left ( 270^\circ +\alpha \right )}=\sin{\alpha }\\[12pt] &\text{tg}{\left ( 270^\circ +\alpha \right )}=-\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 270^\circ +\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\sin{\left ( 270^\circ -\alpha \right )}=-\cos{\alpha }\\[12pt] &\cos{\left ( 270^\circ -\alpha \right )}=-\sin{\alpha }\\[12pt] &\text{tg}{\left ( 270^\circ -\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\text{ctg}{\left ( 270^\circ -\alpha \right )}=\text{tg}{\alpha }\\[22pt] &\sin{\left ( 360^\circ +\alpha \right )}=\sin{\alpha }\\[12pt] &\cos{\left ( 360^\circ +\alpha \right )}=\cos{\alpha }\\[12pt] &\text{tg}{\left ( 360^\circ +\alpha \right )}=\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 360^\circ +\alpha \right )}=\text{ctg}{\alpha }\\[12pt] &\sin{\left ( 360^\circ -\alpha \right )}=-\sin{\alpha }\\[12pt] &\cos{\left ( 360^\circ -\alpha \right )}=\cos{\alpha }\\[12pt] &\text{tg}{\left ( 360^\circ -\alpha \right )}=-\text{tg}{\alpha }\\[12pt] &\text{ctg}{\left ( 360^\circ -\alpha \right )}=-\text{ctg}{\alpha }\\[12pt] \end{split}\]
Sumy i różnice funkcji trygonometrycznych
\[ \sin{\alpha }+\sin{\beta }=2\sin{\frac{\alpha +\beta }{2}}\cos{\frac{\alpha -\beta }{2}}\\[12pt] \sin{\alpha }-\sin{\beta }=2\cos{\frac{\alpha +\beta }{2}}\sin{\frac{\alpha -\beta }{2}}\\[12pt] \cos{\alpha }+\cos{\beta }=2\cos{\frac{\alpha +\beta }{2}}\cos{\frac{\alpha -\beta }{2}}\\[12pt] \cos{\alpha }-\cos{\beta }=-2\sin{\frac{\alpha +\beta }{2}}\sin{\frac{\alpha -\beta }{2}}\\[12pt] \text{tg}{\alpha }+\text{tg}{\beta }=\frac{\sin{\left ( \alpha +\beta \right )}}{\cos{\alpha }\cos{\beta }}\\[12pt] \text{tg}{\alpha }-\text{tg}{\beta }=\frac{\sin{\left ( \alpha -\beta \right )}}{\cos{\alpha }\cos{\beta }}\\[12pt] \text{ctg}{\alpha }+\text{ctg}{\beta }=\frac{\sin{\left ( \beta +\alpha \right )}}{\sin{\alpha }\sin{\beta }}\\[12pt] \text{ctg}{\alpha }-\text{ctg}{\beta }=\frac{\sin{\left ( \beta -\alpha \right )}}{\sin{\alpha }\sin{\beta }} \]
\[ \cos{\alpha }+\sin{\alpha }=\sqrt{2}\sin{\left ( 45^\circ +\alpha \right )}=\sqrt{2}\cos{\left ( 45^\circ -\alpha \right )}\\[12pt] \cos{\alpha }-\sin{\alpha }=\sqrt{2}\cos{\left ( 45^\circ +\alpha \right )}=\sqrt{2}\sin{\left ( 45^\circ -\alpha \right )} \]
\[ \!\!\!\cos{\!\alpha }\!+\!\sin{\!\alpha }\!=\!\sqrt{2}\sin{\!\left ( 45^\circ\!\!\! +\!\alpha \right )}\!=\!\sqrt{2}\cos{\!\left ( 45^\circ\!\!\! -\!\alpha \right )}\\[12pt] \!\!\!\cos{\!\alpha }\!-\!\sin{\!\alpha }\!=\!\sqrt{2}\cos{\!\left ( 45^\circ\!\!\! +\!\alpha \right )}\!=\!\sqrt{2}\sin{\!\left ( 45^\circ\!\!\! -\!\alpha \right )} \]
Sumy i różnice jedności z funkcjami trygonometrycznymi
\[1+\sin{\alpha }=2\sin^2{\left ( 45^\circ +\frac{\alpha }{2} \right )}=2\cos^2{\left ( 45^\circ -\frac{\alpha }{2} \right )}\\[12pt] 1-\sin{\alpha }=2\sin^2{\left ( 45^\circ -\frac{\alpha }{2} \right )}=2\cos^2{\left ( 45^\circ +\frac{\alpha }{2} \right )}\]
\[\!1\!+\!\sin{\alpha }\!=\!2\sin^2\!\!{\left ( 45^\circ\!\! +\!\frac{\alpha }{2} \right )}\!=\!2\cos^2\!\!{\left ( 45^\circ \!\!-\!\frac{\alpha }{2} \right )}\\[12pt] \!1\!-\!\sin{\alpha }\!=\!2\sin^2\!\!{\left ( 45^\circ\!\! -\!\frac{\alpha }{2} \right )}\!=\!2\cos^2\!\!{\left ( 45^\circ \!\!+\!\frac{\alpha }{2} \right )}\]
\[ 1+\cos{\alpha }=2\cos^2{\frac{\alpha }{2}}\\[12pt] 1-\cos{\alpha }=2\sin^2{\frac{\alpha }{2}}\\[12pt] 1+\text{tg}^2{\alpha }=\frac{1}{\cos^2{\alpha }}\\[12pt] 1+\text{ctg}^2{\alpha }=\frac{1}{\sin^2{\alpha }} \]
Różnice kwadratów funkcji trygonometrycznych
\[ \sin^2{\alpha }-\sin^2{\beta }=\cos^2{\beta }-\cos^2{\alpha }=\sin{\left ( \alpha +\beta \right )}\sin{\left ( \alpha -\beta \right )}\\[12pt] \cos^2{\alpha }-\sin^2{\beta }=\cos^2{\beta }-\sin^2{\alpha }=\cos{\left ( \alpha +\beta \right )}\cos{\left ( \alpha -\beta \right )} \]
\[ \!\!\sin^2\!\!{\alpha }\!-\!\sin^2\!\!{\beta }\!=\!\cos^2\!\!{\beta }\!-\!\cos^2\!\!{\alpha }\!=\!\sin{\left ( \alpha\!+\!\beta \right )}\sin{\left ( \alpha\! -\!\beta \right )}\\[12pt] \!\!\cos^2\!\!{\alpha }\!-\!\sin^2\!\!{\beta }\!=\!\cos^2\!\!{\beta }\!-\!\sin^2\!\!{\alpha }\!=\!\cos{\left ( \alpha\!+\!\beta \right )}\cos{\left ( \alpha\! -\!\beta \right )} \]
Iloczyny funkcji trygonometrycznych
\[\begin{split} &\sin{\alpha }\sin{\beta }=\frac{1}{2}\left [ \cos{\left ( \alpha -\beta \right )-\cos{\left ( \alpha +\beta \right )}} \right ]\\[12pt] &\cos{\alpha }\cos{\beta }=\frac{1}{2}\left [ \cos{\left ( \alpha -\beta \right )+\cos{\left ( \alpha +\beta \right )}} \right ]\\[12pt] &\sin{\alpha }\cos{\beta }=\frac{1}{2}\left [ \sin{\left ( \alpha -\beta \right )+\sin{\left ( \alpha +\beta \right )}} \right ]\\[12pt] \end{split}\]