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