Chapter 2: Q38P (page 71)

The functions $s i n t, c o s t, 路路路$ ,are called 鈥渃ircular functions鈥� and the functions $\sinh t, \cosh t, 路路路$ , are called 鈥渉yperbolic functions鈥�. To see a reason for this, show that $x = \cos t, y = \sin t$ , satisfy the equation of a circle $x^{2} + y^{2} = 1$ , while $x = \cosh t, y = \sinh t$ , satisfy the equation of a hyperbola
$x^{2} - y^{2} = 1$ .

Short Answer

Expert verified

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 by step solution

Given Information.

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

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.

Substitute the values in the equation of the circle.

Put

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

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

Substitute the values.

$x^{2} + y^{2} = \cos^{2} (t) + \sin^{2} (t) = 1$

Therefore, it satisfies the equation of the circle.

Substitute the values in the equation of the hyperbola.

Put

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

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

Substitute the values.

$x^{2} + y^{2} = \cosh^{2} (t) + \sinh^{2} (t) = 1$

Therefore, it satisfies the equation of the hyperbola

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Short Answer

Step by step solution

Given Information.

Meaning of rectangular form.

Substitute the values in the equation of the circle.

Substitute the values in the equation of the hyperbola.

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