Q38P - Mathematical Methods in Physical Sciences

Question

The functions $\mathit{s}\mathit{i}\mathit{n}\mathit{t}\mathbf{,}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$, are called “circular functions” and the functions $\mathbf{sinh}\mathbf{ }\mathit{t}\mathbf{,}\mathbf{cosh}\mathbf{ }\mathit{t}\mathbf{,}\mathbf{·}\mathbf{·}\mathbf{·}$ , are called “hyperbolic functions”. To see a reason for this, show that $\mathit{x}\mathbf{=}\mathbf{cos}\mathbf{ }\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sin}\mathbf{ }\mathit{t}$ , satisfy the equation of a circle ${\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$ , while $\mathit{x}\mathbf{=}\mathbf{cosh}\mathbf{ }\mathit{t}\mathbf{,}\mathit{y}\mathbf{=}\mathbf{sinh}\mathbf{ }\mathit{t}$ , satisfy the equation of a hyperbola 

${\mathit{x}}^{\mathbf{2}}\mathbf{-}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{1}$.

Step-by-Step Solution

Answer

It has been proved that the $x=\cos t,y=\sin t$ satisfies the equation of the circle and $x=\cos t,y=\sinh t$ satisfies the equation of the hyperbola.

Step 1: Given Information.

The given equations are, $x=\cos t,y=\sin t$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of in which is the real part and is the imaginary part.

Step 3: Substitute the values in the equation of the circle.

Put

$\begin{array}{l}x=\cos \left(t\right)\\ y=\sin \left(t\right)\end{array}$

 

The equation of the circle is $x^2+y^2=1$.

 

Substitute the values.

$$\begin{gathered} x^2+y^2={\cos }^2\left(t\right)+{\sin }^2\left(t\right) \\ =1 \end{gathered}$$

 

Therefore, it satisfies the equation of the circle.

Step 4: Substitute the values in the equation of the hyperbola.

Put

$\begin{array}{l}x=\cosh \left(t\right)\\ y=\sinh \left(t\right)\end{array}$

 

The equation of the circle is $x^2-y^2=1$.

 

Substitute the values.

$$\begin{gathered} x^2+y^2={\cosh }^2\left(t\right)+{\sinh }^2\left(t\right) \\ =1 \end{gathered}$$

 

Therefore, it satisfies the equation of the hyperbola