Question

What is the exact value of \[\sec \dfrac{\pi }{4}\] ?

Answer 1

Hint:This is a trigonometry function related question. If we know the exact value of the sec function directly we can write the answer. But if not then we can take the help from the basic three functions and those are sin, cos and tan.

Complete step by step answer:
Given is the function \[\sec \dfrac{\pi }{4}\]
We know that, pi is nothing but \[\pi = {180^ \circ }\]
So we can write, \[\sec \dfrac{{{{180}^ \circ }}}{4} = \sec {45^ \circ }\]
But we know that, \[\sec \theta = \dfrac{1}{{\cos \theta }}\]
Thus we can write, \[\sec \dfrac{\pi }{4} = \dfrac{1}{{\cos \dfrac{\pi }{4}}}\]
But we can say, \[\cos {45^ \circ } = \cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\]
So putting this value we get,
\[\sec \dfrac{\pi }{4} = \dfrac{1}{{\dfrac{1}{{\sqrt 2 }}}}\]
The denominator of the denominator will be the numerator,
\[\sec \dfrac{\pi }{4} = \sqrt 2 \]

Hence, the exact value of \[\sec \dfrac{\pi }{4}\] is $\sqrt 2$.

Note:the angle can be given in radians or in degrees. We know that radians to degrees can be done by \[{\theta ^ \circ } \times \dfrac{\pi }{{{{180}^ \circ }}}\]. We either can find the value in any of these types. Note that the second last step is important because if we miss or wrongly write this step the answer will be wrong. So the reciprocals or inverses of the respective functions should be known to us. Like sec and cos, sin and cosec, tan and cot are reciprocals of each other.