According to the trigonometry, the cosecant of zero degrees is equal to infinity.
$\csc{(0^°)} \,=\, \infty$
The exact value of cosecant of zero degrees can be derived in three distinct mathematical approaches. The two of three methods are related to geometry and the remaining method is related to trigonometry. Now, it is your time to learn how to prove the value for cosecant of angle zero radian by each method.
In this geometric method, a property between hypotenuse and opposite side in a zero degree right triangle is considered as a factor for proving the exact value of cosecant of angle zero radian theoretically. In order to prove the cosecant of angle zero grades geometrically, imagine a right triangle that contains zero degrees as its angle. In this case, $\Delta QPR$ is an example triangle, which represents a right angled triangle with zero angle.
According to the definition of the cosecant function, it can be written in mathematical form by writing it in ratio form of lengths of the respective sides. The cosecant function is written as $\csc{(0^°)}$ or $\operatorname{cosec}{(0^°)}$ when the angle of a right triangle equals to zero degrees.
$\implies$ $\csc{(0^°)}$ $\,=\,$ $\dfrac{Length \, of \, Hypotenuse}{Length \, of \, Opposite \, side}$
$\implies$ $\csc{(0^°)}$ $\,=\,$ $\dfrac{PQ}{QR}$
The length of opposite side is zero in a zero degree right triangle. It means $QR \,=\, 0$.
$\require{cancel} \implies$ $\csc{(0^°)}$ $\,=\,$ $\dfrac{d}{0}$
$\,\,\,\therefore\,\,\,\,\,\,$ $\csc{(0^°)}$ $\,=\,$ $\infty$
Practically, the cosecant of angle zero degrees can also be derived by constructing a zero degree right triangle. In fact, it is constructed geometrically by the geometric tools. So, let’s learn the process to construct a right angled triangle with zero degrees angle.
In $\Delta TSU$, the hypotenuse, opposite side and adjacent side are $\overline{ST}$, $\overline{TU}$ and $\overline{SU}$ respectively. Here, $ST = 7\,cm$, $TU = 0\,cm$ and $SU = 7\,cm$. It’s time to find the cosecant of angle zero degrees value by calculating the ratio of lengths of its corresponding sides.
$\csc{(0^°)} \,=\, \dfrac{ST}{TU}$
$\implies$ $\csc{(0^°)} \,=\, \dfrac{7}{0}$
$\,\,\,\therefore\,\,\,\,\,\,$ $\csc{(0^°)} \,=\, \infty$
The exact value for the cosecant of angle zero degrees can also be derived from the fundamentals of trigonometric mathematics. According to the reciprocal trigonometric identity, the cosecant function can be expressed as reciprocal of the sine function.
$\csc{\theta} \,=\, \dfrac{1}{\sin{\theta}}$
The angle $\theta$ is equal to zero degrees in this case.
$\implies$ $\csc{(0^°)} \,=\, \dfrac{1}{\sin{(0^°)}}$
In the trigonometry, we know that the value of sin of zero degrees is equal to zero.
$\implies$ $\csc{(0^°)} \,=\, \dfrac{1}{0}$
$\,\,\,\therefore\,\,\,\,\,\,$ $\csc{(0^°)} \,=\, \infty$
The three methods are possible mathematical methods for proving the exact value of the cosecant of angle zero degrees in mathematics.
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