$\csc{(0^°)}$ Proof

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.

Fundamental 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 $QR \,=\, 0$.

$\require{cancel} \implies$ $\csc{(0^°)}$ $\,=\,$ $\dfrac{d}{0}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\csc{(0^°)}$ $\,=\,$ $\infty$

Experimental method

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.

1. Mark a point $S$ in the plane and draw a horizontal line from that point by a ruler.
2. For drawing a zero angle line, coincide the middle point of protractor with point $S$ and also coincide its right side baseline with horizontal line. Now, draw a straight line with zero degree angle from point $S$. The zero angle line is drawn actually over the horizontal line.
3. Now, take compass and then set the distance between pencil lead and needle points to any length (for example $7$ centimetres). After that, draw an arc on zero degrees line from point $S$ and it intersects the zero angle straight line at point $T$.
4. Lastly, from point $T$ draw a perpendicular line to horizontal line by a set square. It intersects the horizontal line at point $U$. In this way, we construct $\Delta TSU$ geometrically. Remember that it is complicated to draw a perpendicular line to horizontal line and it is impossible due to the same location of the points $T$ and $U$ in the plane.

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.

$\csc{(0^°)} \,=\, \dfrac{ST}{TU}$

$\implies$ $\csc{(0^°)} \,=\, \dfrac{7}{0}$

$\,\,\,\therefore\,\,\,\,\,\,$ $\csc{(0^°)} \,=\, \infty$

Trigonometric method

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.