Question

How do you evaluate \[\sec (\pi )\]?

Answer 1

Hint:
Real functions which relate any angle of a right angled triangle to the ratio of any two of its sides are called trigonometric functions. We can also use geometric definitions to evaluate trigonometric values. It is known that the inverse of cosine of theta is equivalent to the secant of theta. Here, it’s important that we know the cosine of theta is the ratio of the adjacent side (base) to the hypotenuse.

Complete step by step solution:
According to the given data, we need to evaluate \[\sec (\pi )\]
If in a right angled triangle \[\theta \] represents one of its acute angle then by definition we can write
\[\cos \theta = \dfrac{{Base}}{{Hypotenuse}}\]
Thereafter we know that,
\[\sec \theta = \dfrac{1}{{\cos \theta }} = \dfrac{{Base}}{{Hypotenuse}}\]
Therefore, we directly use the inverse formula to evaluate the value of secant of theta.
\[\sec \theta = \dfrac{1}{{\cos \theta }}\]
According to the given data, \[\theta = \pi \].
Hence,
\[ \Rightarrow \sec (\pi ) = \dfrac{1}{{\cos (\pi )}}\]
Now when we use this trivial identity:
\[\cos (\pi ) = - 1\]
When we substitute the value in the expression, we get
\[ \Rightarrow \sec (\pi ) = \dfrac{1}{{\cos (\pi )}} = \dfrac{1}{{ - 1}} = - 1\]
This gives rise to the fact that \[\sec (\pi ) = - 1\]

Hence, the value of \[\sec (\pi )\] is equivalent to \[ - 1\].

Note:
Trigonometric functions are real functions which relate any angle of a right angled triangle to the ratio of any two of its sides. One should be careful while evaluating trigonometric values and rearranging the terms to convert from one function to the other. The widely used ones are sin, cos and tan. While the rest can be referred to as the inverse of the other trigonometric ratios, i.e., cosec, sec and cot respectively. If in a right angled triangle θ represents one of its acute angles then, \[\sec \theta = \dfrac{1}{{\cos \theta }}\].