Формулы. Математика. Тригонометрия
Полезные тождества (2)
Полезные тождества (2)
\[
\sin^{2}x
\]
\[
\cos^{2}x
\]
\[
\sin^4(x)+\cos^4(x)
\]
\[
\sin^6(x)+\cos^6(x)
\]
\[
1\pm \mathtt{tg} x
\]
\[
1\pm \mathtt{ctg} x
\]
\[
\frac{\sin 2x}{\cos 2x +1}
\]
\[
\sin \left (\frac {\pi}{4}+x \right)
\]
\[
\sin \left (\frac {\pi}{4}-x \right)
\]
\[
1\pm \sin x
\]
\[
1+\cos x
\]
\[
1-\cos x
\]
\[
\sin 3x
\]
\[
\mathtt{tg} 3x
\]
\[
\sin 5x
\]
\[
\mathtt{tg} 5x
\]
\[
\sin 7x
\]
\[
\mathtt{tg} 7x
\]
\[
\cos(20^o) \cdot \cos(40^o) \cdot \cos(80^o)
\]
\[
\cos \left (\frac{\pi}{7} \right) \cdot \cos \left (\frac{4\pi}{7}\right) \cdot \cos \left (\frac{5\pi}{7}\right)
\]
\[
\cos \left (\frac{\pi}{7} \right) \cdot \cos \left (\frac{2\pi}{7}\right) \cdot \cos \left (\frac{4\pi}{7}\right)
\]
\[
\cos \left (\frac{\pi}{9}\right) \cdot \cos \left (\frac{2\pi}{9}\right) \cdot \cos \left (\frac{4\pi}{9}\right)
\]
\[
\cos \left (\frac{\pi}{9}\right) \cdot \cos \left (\frac{5\pi}{9}\right) \cdot \cos \left (\frac{7\pi}{9}\right)
\]
\[
\mathtt{arctg}\left (\frac 12 \right )+\mathtt{arctg}\left (\frac 13 \right)
\]
\[
\sin^{2}x = {\frac{1}{1+\mathtt{ctg}^{2}x}}
\]
\[
\cos^{2}x = \frac{1}{1+\mathtt{tg}^2 x}
\]
\[
\sin^4(x)+\cos^4(x) = 1-2 \sin^2(x)\, \cos^2(x) = 1-\frac 12 \sin^2(2x) = \frac 34 +\frac 14 \cos(4x)
\]
\[
\sin^6(x)+\cos^6(x) = 1-3 \sin^2(x)\, \cos^2(x) = 1-3 \sin^2(x)+3sin^4(x) = 1-\frac 34 \sin^2(2x) = \frac 58+\frac 38 \cos(4x)
\]
\[
1\pm \mathtt{tg} x = \frac{\sqrt{2} \sin \left (\frac{\pi}{4}\pm x \right )}{\cos x }
\]
\[
1\pm \mathtt{ctg} x = \frac{\sqrt{2} \sin \left (\frac{\pi}{4}\pm x \right )}{\sin x }
\]
\[
\frac{\sin 2x}{\cos 2x +1} = \frac{1- \cos 2x }{\sin 2x} = \mathtt{tg} x
\]
\[
\sin \left (\frac {\pi}{4}+x \right) = \cos \left (\frac {\pi}{4}-x \right)
\]
\[
\sin \left (\frac {\pi}{4}-x \right) = \cos \left (\frac {\pi}{4}+x \right)
\]
\[
1\pm \sin x = 2 \sin^2 \left (\frac {\pi}{4} \pm \frac x2 \right)
\]
\[
1+\cos x = 2 \cos^2 \left (\frac x2 \right)
\]
\[
1-\cos x = 2 \sin^2 \left (\frac x2 \right)
\]
\[
\sin 3x = 4 \sin x \cdot \sin \left(\frac{\pi}{3}+x\right) \cdot \sin \left (\frac{pi}{3}-x\right)
\]
\[
\mathtt{tg} 3x = \mathtt{tg} x \cdot \mathtt{tg}\left(\frac{\pi}{3}+x\right )\cdot \mathtt{tg}\left(\frac{\pi}{3}-x\right)
\]
\[
\sin 5x = 16 \sin x \cdot \sin \left(\frac{\pi}{5}+x\right)\cdot \sin \left (\frac{pi}{5}-x\right) \cdot \sin \left(\frac{2\pi}{5}+x\right)\cdot \sin \left(\frac{2\pi}{5}-x\right)
\]
\[
\mathtt{tg} 5x = \mathtt{tg} x \cdot \mathtt{tg}\left(\frac{\pi}{5}+x\right)\cdot \mathtt{tg}\left(\frac{\pi}{5}-x\right) \cdot \mathtt{tg}\left (\frac{2\pi}{5}+x\right)\cdot \mathtt{tg}\left (\frac{2\pi}{5}-x\right)
\]
\[
\sin 7x = 64 \sin x \cdot \sin\left (\frac{\pi}{7}+x\right)\cdot \sin \left (\frac{\pi}{7}-x\right) \cdot \sin \left (\frac{2\pi}{7}+x\right)\cdot \sin \left (\frac{2\pi}{7}-x\right) \cdot \sin \left (\frac{3\pi}{7}+x\right)\cdot \sin \left (\frac{3\pi}{7}-x\right)
\]
\[
\mathtt{tg} 7x = \mathtt{tg} x \cdot \mathtt{tg}\left (\frac{\pi}{7}+x\right)\cdot \mathtt{tg}\left (\frac{\pi}{7}-x\right) \cdot \mathtt{tg}\left (\frac{2\pi}{7}+x\right)\cdot \mathtt{tg}\left (\frac{2\pi}{7}-x\right) \cdot \mathtt{tg}\left (\frac{3\pi}{7}+x\right)\cdot \mathtt{tg}\left (\frac{3\pi}{7}-x\right)
\]
\[
\cos(20^o) \cdot \cos(40^o) \cdot \cos(80^o) = \frac 18
\]
\[
\cos \left (\frac{\pi}{7} \right) \cdot \cos \left (\frac{4\pi}{7}\right) \cdot \cos \left (\frac{5\pi}{7}\right) = \frac 18
\]
\[
\cos \left (\frac{\pi}{7} \right) \cdot \cos \left (\frac{2\pi}{7}\right) \cdot \cos \left (\frac{4\pi}{7}\right) = -\frac 18
\]
\[
\cos \left (\frac{\pi}{9}\right) \cdot \cos \left (\frac{2\pi}{9}\right) \cdot \cos \left (\frac{4\pi}{9}\right) = \frac 18
\]
\[
\cos \left (\frac{\pi}{9}\right) \cdot \cos \left (\frac{5\pi}{9}\right) \cdot \cos \left (\frac{7\pi}{9}\right) = \frac 18
\]
\[
\mathtt{arctg}\left (\frac 12 \right )+\mathtt{arctg}\left (\frac 13 \right) = \frac{\pi}{4}
\]
