Question

Why is \[\sec \left( 0 \right) = 1\]?

Answer 1

Hint: Here, in the given question, we are asked to state the reason for \[\sec \left( 0 \right)\] being equal to \[1\]. We will first understand the meaning of secant of an angle, its relation with cosecant of an angle and then we will be able to state the reason of \[\sec \left( 0 \right)\] being equal to \[1\].

Complete step-by-step solution:
Meaning of \[\sec \left( 0 \right)\]: The ratio of the length of the hypotenuse to the length of the adjacent side (usually base of a right angle triangle) is the secant of an angle formed by these two sides.
Also, we know that the ratio of length of the adjacent side to the length of hypotenuse is the cosecant of an angle formed by these two sides.
We can observe that, by meaning of secant and cosecant, both the functions are exactly opposite to each other. Therefore, we can use cosecant in place of secant in the form: \[\text{Secant} = \dfrac{1}{\text{Cosecant}}\].
Since, the value of \[\cos \left( 0 \right)\] is equal to \[1\],
Therefore, \[\sec \left( 0 \right) = \dfrac{1}{{\cos \left( 0 \right)}}\]
\[ \Rightarrow \sec \left( 0 \right) = \dfrac{1}{1}\]
And hence, \[\sec \left( 0 \right) = 1\]

Note: Let us understand the deeper meaning of secant of an angle measuring zero degree. As we have discussed it is the ratio of the length of the hypotenuse to the length of the adjacent side. And its numerical value is one as we have calculated above. This means that the length of the hypotenuse and the adjacent side is equal to one another, which is not practically possible in a right-angled triangle as it will not satisfy Pythagoras theorem. Also angle measuring zero degree means the hypotenuse and adjacent side are not two different sides.