Question

How do you find the exact values of cot, cosec and sec for 180 degrees?

Answer 1

Hint: We will first of all, write 180 in terms of 90 + 90 and then we will just use the identities we have with us regarding the cot, cosec and sec of 90 + x to get the required answer.

Complete step by step solution:
We are given that we are required to find the exact values of cot, cosec and sec for 180 degrees.
Since, we can write 180 as the sum of 90 and 90.
Therefore, we can write the cotangent of 180 degrees as follows:-
$ \Rightarrow \cot {180^ \circ } = \cot ({90^ \circ } + {90^ \circ })$
Now, since we know that we have a formula given by the following expression with us:-
$ \Rightarrow \cot ({90^ \circ } + \theta ) = - \tan \theta $
Replacing $\theta $ by ${90^ \circ }$ in the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow \cot {180^ \circ } = - \tan {90^ \circ }$
Now, since we also know that $\tan {90^ \circ } = \infty $, therefore, we have the following expression with us:-
$ \Rightarrow \cot {180^ \circ } = - \infty $
Now, we can write the cosecant of 180 degrees as follows:-
$ \Rightarrow \csc {180^ \circ } = \csc ({90^ \circ } + {90^ \circ })$
Now, since we know that we have a formula given by the following expression with us:-
$ \Rightarrow \csc ({90^ \circ } + \theta ) = \sec \theta $
Replacing $\theta $ by ${90^ \circ }$ in the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow \csc {180^ \circ } = \sec {90^ \circ }$
Now, since we also know that $\sec {90^ \circ } = \infty $, therefore, we have the following expression with us:-
$ \Rightarrow \csc {180^ \circ } = \infty $
Now, we can write the secant of 180 degrees as follows:-
$ \Rightarrow \sec {180^ \circ } = \sec ({90^ \circ } + {90^ \circ })$
Now, since we know that we have a formula given by the following expression with us:-
$ \Rightarrow \sec ({90^ \circ } + \theta ) = - \csc \theta $
Replacing $\theta $ by ${90^ \circ }$ in the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow \sec {180^ \circ } = \csc {90^ \circ }$
Now, since we also know that $\csc {90^ \circ } = 1$, therefore, we have the following expression with us:-
$ \Rightarrow \sec {180^ \circ } = 1$

Note: The students must note that we have a phrase given by “ADD SUGAR TO COFFEE”, which states that all the trigonometric ratios are positive in first quadrant, since and cosecant are positive in second quadrant, tangent and cotangent are positive in second quadrant and cosine and secant are positive in fourth quadrant.
Therefore, when we added some angle to 90 degrees, we reached the second quadrant and only since cosecant are positive in that quadrant.