Wzory trygonometryczne

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

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