\[\cases{{}\\ Cos\ -φ=Cos\ φ=Cos\left({π±φ}\right)\\ {}\\ Sin\ -φ=-Sin\ φ=Sin\left({π+φ}\right)\\ {}\\ \qquad \qquad \qquad Sin\ \mathbf{φ}=Sin\left({π-\mathbf{φ}}\right)\\ {}\\ Cos\ φ=Sin\left({φ+\frac π2}\right)\ ,\ Cos\frac π4=Sin\frac{3π}4=Sin\left({π-\mathbf{\frac π4}}\right)=Sin\mathbf{\frac π4}\\ {}\\ Sin\ Θ=Cos\left({Θ-\frac π2}\right)\ ,\ Sin\frac π4=Cos\left({-\frac π4}\right)=Cos\frac π4\\ {}} \]
\[\displaystyle{Sin\ x=\sum_{i=1}^n\left({-1}\right)^{n-1}\frac{x^{2n-1}}{\left({2n-1}\right)!}}\]
\[\displaystyle{Cos\ x=1-\sum_{i=1}^n\left({-1}\right)^{n-1}\frac{x^{2n}}{\left({2n}\right)!}}\]
\[Sin\left({α±ß}\right)=Sin\ α·Cos\ ß±Cos\ α·Sin\ ß\]
\[Cos\left({α±ß}\right)=Cos\ α·Cos\ ß∓Sin\ α·Sin\ ß\]
\[Tan\left({α±ß}\right)=\frac{Tan\ α±Tan\ ß}{1∓Tan\ α·Tan\ ß}\]
\[Cot\left({α±ß}\right)=\frac{Cot\ \mathbf{ß}·Cot\ \mathbf{α}∓1}{Cot\ \mathbf{ß}±Cot\ \mathbf{α}}\]
\[Sin\ α±Sin\ ß=2·Sin\frac{α±ß}2·Cos\frac{α∓ß}2\]
\[Cos\ α±Cos\ ß=2·{}^{Cos}_{Sin}\frac{α+ß}2·{}^{Cos}_{Sin}\frac{α-ß}2\]
\[Tan\ α±Tan\ ß=\frac{Sin\left({α±ß}\right)}{Cos\ α·Cos\ ß}\]
\[Cot\ α±Cot\ ß=\frac{Sin\left({\mathbf{ß}±\mathbf{α}}\right)}{Sin\ \mathbf{ß}·Sin\ \mathbf{α}}\]
\[Sin^2\ α+Cos^2\ α=1=\\ =\left({Cos\ α-i·Sin\ α}\right)\left({Cos\ α+i·Sin\ α}\right)=\\ =\left({ch\ iα-sh\ iα}\right)\left({ch\ iα+sh\ iα}\right)=\\ =ch^2\ φ-sh^2\ φ\]
\[Tan^k\ α=Cot^{-k}\ α=\frac{Sin^k\ α}{Cos^k\ α}\]
\[Sin\left({2α}\right)=2·Sin\ α·Cos\ α\]
\[Cos\left({2α}\right)=Cos^2\ α-Sin^2\ α=\\ =2·Cos^2\ α-1=1-2·Sin^2\ α\]
\[Tan\left({2α}\right)=\frac{2·Tan\ α}{1-Tan^2\ α}\]
\[Cot\left({2α}\right)=\frac{Cot^2\ α-1}{2·Cot\ α}\]
\[1-Cos\ α=2·Sin^2\frac α2\]
\[1+Cos\ α=2·Cos^2\frac α2\]
\[1-Sin\ α=\left({Cos\frac α2-Sin\frac α2}\right)^2\]
\[1+Sin\ α=\left({Cos\frac α2+Sin\frac α2}\right)^2\]
\[Sin\frac α2=±\sqrt{\frac{1-Cos\ α}2}\]
\[Cos\frac α2=±\sqrt{\frac{1+Cos\ α}2}\]
\[Tan\frac α2=±\sqrt{\frac{1-Cos\ α}{1+Cos\ α}}=\frac{1-Cos\ α}{Sin\ α}=\frac{Sin\ α}{1+Cos\ α}\]
\[1+Tan^2\ α=\frac1{Cos^2\ α}\]
\[1+Cot^2\ α=\frac1{Sin^2\ α}\]
\[Sec\ α=\frac1{Cos\ α}=±\sqrt{1+Tan^2\ α}\]
\[Cosec\ α=\frac1{Sin\ α}=±\sqrt{1+Cot^2\ α}\]
| . | \(Sin\ α\) | \(Cos\ α\) | \(Tan\ α\) | \(Cot\ α\) |
|---|---|---|---|---|
| \(Sin\ α\) | . | \(\left({1-Cos^2\ α}\right)^{\frac12}\) | \(\left({Tan^{-2}\ α+1}\right)^{-\frac12}\) | \(\left({1+Cot^2\ α}\right)^{-\frac12}\). |
| \(Cos\ α\) | \(\left({1-Sin^2\ α}\right)^{\frac12}\) | . | \(\left({1+Tan^2\ α}\right)^{-\frac12}\) | \(\left({1+Cot^{-2}\ α}\right)^{-\frac12}\) |
| \(Tan\ α\) | \(\left({Sin^{-2}\ α-1}\right)^{-\frac12}\) | \(\left({Cos^2\ α-1}\right)^{\frac12}\) | . | \(Cot^{-1}\ α\) |
| \(Cot\ α\) | \(\left({Sin^{-2}\ α-1}\right)^{\frac12}\) | \(\left({Cos^2\ α-1}\right)^{-\frac12}\) | \(Tan^{-1}\ α\) | . |
| \(Cos\ φ\) | \(0\) | \(-1\) | \(0\) | \(1\) | \(Cos\ φ\) | \(\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}2\) | \(\frac{\sqrt{2+\sqrt{3}}}2\) | \(\frac{\sqrt{2+\sqrt{2}}}2\) | \(\frac{\sqrt{3}}2\) | \(\frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}2\) | \(\frac{\sqrt{2}}2\) | \(\frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}2\) | \(\frac 12\) | \(\frac{\sqrt{2-\sqrt{2}}}2\) | \(\frac{\sqrt{2-\sqrt{3}}}2\) | \(\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}2\) | \(Cos\ φ\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(Sin\ φ\) | \(-1\) | \(0\) | \(1\) | \(0\) | \(Sin\ φ\) | \(\frac{\sqrt{2-\sqrt{2+\sqrt{3}}}}2\) | \(\frac{\sqrt{2-\sqrt{3}}}2\) | \(\frac{\sqrt{2-\sqrt{2}}}2\) | \(\frac 12\) | \(\frac{\sqrt{2-\sqrt{2-\sqrt{3}}}}2\) | \(\frac{\sqrt{2}}2\) | \(\frac{\sqrt{2+\sqrt{2-\sqrt{3}}}}2\) | \(\frac{\sqrt{3}}2\) | \(\frac{\sqrt{2+\sqrt{2}}}2\) | \(\frac{\sqrt{2+\sqrt{3}}}2\) | \(\frac{\sqrt{2+\sqrt{2+\sqrt{3}}}}2\) | \(Sin\ φ\) |
| \(φ\) | \(\frac{3π}2\) | \(π\) | \(\frac π2\) | \(2kπ\) | \(φ\) | \(\frac{π}{24}\) | \(\frac{π}{12}\) | \(\frac π8\) | \(\frac π6\) | \(\frac{5π}{24}\) | \(\frac π4\) | \(\frac{7π}{24}\) | \(\frac π3\) | \(\frac{9π}{24}\) | \(\frac{5π}{12}\) | \(\frac{11π}{24}\) | \(φ\) |
| \(Tan\ φ\) | \(±∞\) | \(0\) | \(±∞\) | \(0\) | \(Tan\ φ\) | \(\frac{\sqrt{2-\sqrt{3}}}{2+\sqrt{2+\sqrt{3}}}\) | \(2-\sqrt{3}\) | \(\frac1{\sqrt{2}+1}\) | \(\frac1{\sqrt{3}}\) | \(\frac{\sqrt{2+\sqrt{3}}}{2+\sqrt{2-\sqrt{3}}}\) | \(1\) | \(\frac{\sqrt{2+\sqrt{3}}}{2-\sqrt{2-\sqrt{3}}}\) | \(\sqrt{3}\) | \(\frac1{\sqrt{2}-1}\) | \(2+\sqrt{3}\) | \(\frac{\sqrt{2-\sqrt{3}}}{2-\sqrt{2+\sqrt{3}}}\) | \(Tan\ φ\) |
| \(Cot\ φ\) | \(0\) | \(±∞\) | \(0\) | \(±∞\) | \(Cot\ φ\) | \(\frac{\sqrt{2-\sqrt{3}}}{2-\sqrt{2+\sqrt{3}}}\) | \(2+\sqrt{3}\) | \(\frac1{\sqrt{2}-1}\) | \(\sqrt{3}\) | \(\frac{\sqrt{2+\sqrt{3}}}{2-\sqrt{2-\sqrt{3}}}\) | \(1\) | \(\frac{\sqrt{2+\sqrt{3}}}{2+\sqrt{2-\sqrt{3}}}\) | \(\frac1{\sqrt{3}}\) | \(\frac1{\sqrt{2}+1}\) | \(2-\sqrt{3}\) | \(\frac{\sqrt{2-\sqrt{3}}}{2+\sqrt{2+\sqrt{3}}}\) | \(Cot\ φ\) |
\[arcCos\ A=±i·ln\left({A±\sqrt{A^2-1}}\right)\]
\[arcSin\ A=\left({±}\right)\ i·ln\left({±\sqrt{1-A^2}+\left({∓}\right)\ i·A}\right)\]
\[\frac i2\ ln\ \frac{1-i·A}{1+i·A}=arcTan\ A=\frac i2\ ln\ \frac{i+A}{i-A}\]
\[\frac i2\ ln\ \frac{i·A+1}{i·A-1}=arcCot\ A=\frac i2\ ln\ \frac{A-i}{A+i}\]
\[arcTan\ A+arcCot\ A=\frac π2±2kπ\]
\[±\left({arcTan\ A-arcCot\ A}\right)=\frac i2\ ln\ \left({i·\frac{i·A∓1}{i·A±1}}\right)^2=\frac i2\ ln\ \left({-i·\frac{1∓i·A}{1±i·A}}\right)^2=\frac i2\ ln\ \left({i·\frac{A±i}{A∓i}}\right)^2=etc. ...\]
[Eop]
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