5.6.1 Complex Number System 馃敡
Proof of Euler's Formula
Proof of Negative Natural Logarithms
Proof of Imaginary Natural Logarithms
5.6.2 Analytic Function for $\sin(\theta)$ 馃敡
$$\sin(\theta)=\frac{e^{i路\theta}-e^{-i路\theta}}{2路i}$$
Proof
5.6.3 Analytic Function for $\cos(\theta)$ 馃敡
$$\cos(\theta)=\frac{e^{i路\theta}+e^{-i路\theta}}{2路i}$$
Proof
5.6.4 Analytic Function for $\csc(\theta)$ 馃敡
$$\csc(\theta)=\frac{2路i}{e^{i路\theta}-e^{-i路\theta}}$$
Proof
5.6.5 Analytic Function for $\sec(\theta)$ 馃敡
$$\sec(\theta)=\frac{2路i}{e^{i路\theta}+e^{-i路\theta}}$$
Proof
5.6.6 Analytic Function for $\tan(\theta)$ 馃敡
$$\tan(\theta)=\frac{e^{i路\theta}-e^{-i路\theta}}{i路\big(e^{i路\theta}+e^{-i路\theta}\big)}$$
Proof
5.6.7 Analytic Function for $\cot(\theta)$ 馃敡
$$\cot(\theta)=\frac{i路\big(e^{i路\theta}+e^{-i路\theta}\big)}{e^{i路\theta}-e^{-i路\theta}}$$
Proof
5.6.8 Analytic Function for $\sin^{-1}(x)$ 馃敡
$$\sin^{-1}(x)=-i路\ln\big(i路x+\sqrt{1-x^2}\big)$$
Proof
5.6.9 Analytic Function for $\cos^{-1}(x)$ 馃敡
$$\cos^{-1}(x)=-i路\ln\big(x+i路\sqrt{1-x^2}\big)$$
Proof
5.6.A Analytic Function for $\csc^{-1}(x)$ 馃敡
$$\csc^{-1}(x)=-i路\ln\big(i路x^{-1}+\sqrt{1-x^{-2}}\big)$$
Proof
5.6.B Analytic Function for $\sec^{-1}(x)$ 馃敡
$$\sec^{-1}(x)=-i路\ln\big(x^{-1}+i路\sqrt{1-x^{-2}}\big)$$
Proof
5.6.C Analytic Function for $\tan^{-1}(x)$ 馃敡
$$\tan^{-1}(x)=\frac{i}{2}路ln\bigg(\frac{1-i路x}{1+i路x}\bigg)$$
Proof
5.6.D Analytic Function for $\cot^{-1}(x)$ 馃敡
$$\cot^{-1}(x)=\frac{i}{2}路ln\bigg(\frac{i路x+1}{i路x-1}\bigg)$$
Proof
