Zależności funkcji trygonometrycznych

Podstawowe zależności:

(1)
\begin{align} \sin^2\alpha+\cos^2\alpha=1 \end{align}
(2)
\begin{align} tg\alpha=\frac{\sin\alpha}{\cos\alpha} \end{align}
(3)
\begin{align} tg\alpha ctg\alpha=1 \end{align}

(4)
\begin{align} \sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta \end{align}
(5)
\begin{align} \sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta \end{align}
(6)
\begin{align} \sin2\alpha=2\sin\alpha\cos\alpha \end{align}
(7)
\begin{align} \sin3\alpha=3\cos^2\alpha\sin\alpha-sin^3\alpha \end{align}
(8)
\begin{align} \sin4\alpha=4\cos^3\alpha\sin\alpha-4cos\alpha\sin^3\alpha=4\sin\alpha\cos\alpha(\cos^2\alpha-\sin^2\alpha) \end{align}
(9)
\begin{align} \sin5\alpha=5\cos^4\alpha\sin\alpha-10cos^2\alpha\sin^3\alpha+sin^5\alpha \end{align}

(10)
\begin{align} \cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta \end{align}
(11)
\begin{align} \cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta \end{align}
(12)
\begin{align} \cos2\alpha=\cos^2\alpha-\sin^2\alpha=1-2\sin^2\alpha=2\cos^2\alpha-1 \end{align}
(13)
\begin{align} \cos3\alpha=\cos^3\alpha-3\cos\alpha\sin^2\alpha \end{align}
(14)
\begin{align} \cos4\alpha=\cos^4\alpha-6\cos^2\alpha\sin^2\alpha+\sin^4\alpha \end{align}
(15)
\begin{align} \cos5\alpha=\cos^5\alpha-10\cos^3\alpha\sin^2\alpha+5\cos\alpha\sin^4\alpha \end{align}

(16)
\begin{align} tg(\alpha+\beta)=\frac{tg\alpha+tg\beta}{1-tg\alpha tg\beta} \end{align}
(17)
\begin{align} tg(\alpha-\beta)=\frac{tg\alpha-tg\beta}{1+tg\alpha tg\beta} \end{align}
(18)
\begin{align} tg2\alpha=\frac{2 tg\alpha}{1-tg^2\alpha} \end{align}

(19)
\begin{align} tg\alpha=\frac{2tg\frac{\alpha}{2}}{1-tg^2\frac{\alpha}{2}} \end{align}
(20)
\begin{align} ctg\alpha=\frac{1-tg^2\frac{\alpha}{2}}{2tg\frac{\alpha}{2}} \end{align}
(21)
\begin{align} \sin\alpha=\frac{2tg\frac{\alpha}{2}}{1+tg^2\frac{\alpha}{2}} \end{align}
(22)
\begin{align} \cos\alpha=\frac{1-tg^2\frac{\alpha}{2}}{1+tg^2\frac{\alpha}{2}} \end{align}

Uogólnione zależności:

(23)
\begin{align} \sin n\alpha=\displaystyle\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k{2k+1 \choose n}\cos^{n-2k-1}\alpha\sin^{2k+1}\alpha \end{align}
(24)
\begin{align} \cos n\alpha=\displaystyle\sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k{2k \choose n}\cos^{n-2k}\alpha\sin^{2k}\alpha \end{align}
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