According lớn trigonometric mathematics, the cot of zero degrees is equal khổng lồ infinity.

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\$cot(0^°) ,=, infty\$

The cot of angle zero degrees can be derived in three different methods. One of them is trigonometric approach and remaining two are geometric methods. So, let’s learn every mathematical approach for proving that the cot of angle zero degrees is equal khổng lồ infinity.

### Fundamental method

In this method, a geometric property between sides of a zero degree right triangle is considered for deriving the exact value of cotangent of angle zero degrees. Now, let’s study how lớn prove it by imagining a right triangle with zero degrees angle. The \$Delta RPQ\$ is an example for a zero angle right triangle.

According khổng lồ the fundamental definition of cot function, it can be written in ratio of lengths of the sides. The cot function is written as \$cot(0^°)\$ when the angle of a right triangle is zero degrees.

\$implies\$ \$cot(0^°)\$ \$,=,\$ \$dfracLength , of , Adjacent , sideLength , of , Opposite , side\$

\$implies\$ \$cot(0^°)\$ \$,=,\$ \$dfracPRQR\$

In a zero degree right triangle, the length of opposite side is zero. Therefore \$QR ,=, 0\$.

\$implies\$ \$cot(0^°)\$ \$,=,\$ \$dfracd0\$

\$,,, herefore,,,,,,\$ \$cot(0^°)\$ \$,=,\$ \$infty\$

### Experimental method

A right triangle with zero degrees angle is constructed in this method by the geometric tools for proving the cotangent of angle zero degrees practically. Let’s learn the procedure for constructing a right angled triangle with zero degrees angle.

Consider a point \$J\$ & draw a horizontal line from this point by a ruler.Now, take protractor và draw a zero degree line by coinciding the middle point with point \$J\$ và also coinciding its right side baseline with horizontal line. In fact, the zero degrees line is drawn over the horizontal straight line.After that, mix the distance between needle point & pencil lead point of a compass lớn any length (for example \$6\$ centimetres). Now, draw an arc on zero degrees line and it cuts the zero degrees line at point \$K\$.Lastly, draw a perpendicular line khổng lồ horizontal line from point \$K\$ by a set square. It intersects the horizontal line at point \$L\$. Thus, we construct a \$Delta KJL\$. Remember that it is not possible to lớn draw a perpendicular line lớn horizontal line from point \$K\$ because the points \$K\$ & \$L\$ are at same position in the plane.

In \$Delta KJL\$, the opposite side, adjacent side and hypotenuse are \$overlineKL\$, \$overlineJL\$ & \$overlineJK\$ respectively. \$KL = 0,cm\$, \$JK = 6,cm\$ & \$JL = 6,cm\$. Now, we can calculate the cotangent of angle zero degrees by the ratio of lengths of its corresponding sides.

\$cot(0^°) ,=, dfracJLKL\$

\$implies\$ \$cot(0^°) ,=, dfrac60\$

\$,,, herefore,,,,,,\$ \$cot(0^°) ,=, infty\$

### Trigonometric method

In trigonometry, the cotangent of angle zero degrees can be proved from the fundamentals of trigonometry. According lớn quotient trigonometric identity, the cot function can be expressed in terms of cos and sine function.

\$cot heta ,=, dfraccos hetasin heta\$

In this case, the angle \$ heta\$ is zero degrees.

\$implies\$ \$cot(0^°) ,=, dfraccos(0^°)sin(0^°)\$

We know that the value of cos of zero degrees is one & the value of sine of zero degrees is equal khổng lồ zero. Now, replace them in the above equation for evaluating the cot of \$0\$ degrees.

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\$implies\$ \$cot(0^°) ,=, dfrac10\$

\$,,, herefore,,,,,,\$ \$cot(0^°) ,=, infty\$

These are the possible ways to prove that the co-tangent of angle zero degrees is equal to lớn infinity in mathematics.