Q18P - Mathematical Methods in Physical Sciences

Question

Verify the formulas in Problems 17 to 24.

$\mathit{a}\mathit{r}\mathit{c}\mathit{c}\mathit{o}\mathit{s}\mathit{z}\mathbf{=}\mathit{i}\mathit{l}\mathit{n}\left(z\pm \sqrt{z^2-1}\right)$

Step-by-Step Solution

Answer

The formula$arccosz=iln\left(z\pm \sqrt{z^2-1}\right)$is verified. 

Step 1: Given Information

Given formula is $arccosz=iln\left(z\pm \sqrt{z^2-1}\right)$ .

Step 2: Definition of Trigonometric equation.

An equation involving one or more trigonometric ratios of unknown angles is known as a trigonometric equation.

Step 3: Use exponential form to expand the equation

Given the function $arccosz=iln\left(z\pm \sqrt{z^2-1}\right)$.

Write the exponential form of the $w=arc\cos ( z).$ .

$$\begin{gathered} z=\cos \left(w\right) \\ =\frac{e^{\left(wi\right)}+e^{\left(-wi\right)}}{2} \end{gathered}$$                                                                                                       …(1)

 

Rewrite the equation (1).

  $$\begin{gathered} e^{\left(wi\right)}+e^{\left(-wi\right)}=2z \\ {e}^{\left(2wi\right)}+2z{e}^{\left(wi\right)}+1=0 \end{gathered}$$                                                                                          …(2)

 

Let $u=e^{\left(wi\right)}$in equation (2).

 $u^2-2zu+1=0$                                                                                                       …(3)

 

The coefficient of equation is as follows.

$$\begin{gathered} a=1 \\ b=-2z \\ c=1 \end{gathered}$$

 

Use quadratic formula to find roots of equation (3).

 $$\begin{gathered} u=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ =\frac{2z\pm \sqrt{{\left(-2z\right)}^2-4}}{2} \\ =z\pm \sqrt{z^2-1} \end{gathered}$$

Replace value of u by $u=e^{\left(wi\right)}$  .

$wi=\ln \left(z\pm \sqrt{z^2-1}\right)$

 

Replace value of w by $w=arccos\left(z\right)$ .

$arccos\left(z\right)=i\ln \left(z\pm \sqrt{z^2-1}\right)$

Hence the formula is verified.