Question

How do you find the value of$sec\left( {315} \right)$ ?

Answer 1

Hint : While solving this particular question we must convert the given trigonometric function into its corresponding simpler form that is $\sec (315) = \dfrac{1}{{\cos (315)}}$ that manipulate ${315^ \circ } = {360^ \circ } - {45^ \circ }$ , then apply the even function identity , at the end you will get your desired result . Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.

Complete step-by-step answer :
We have to find the value of given trigonometric ratio that is $sec\left( {315} \right)$ ,
We already know the relationship between secant and cosine that is ,
$\sec x = \dfrac{1}{{\cos x}}$
Therefore, we can write the given expression as ,
$\sec (315) = \dfrac{1}{{\cos (315)}}$
Here $315$ must be in the degree form, therefore , we can write this as ${315^ \circ } = {360^ \circ } - {45^ \circ }$ , and since ${360^ \circ }$ is a complete loop ,
And we know that $\cos ({360^ \circ } - x) = \cos ( - x)$ we will get the following result,
$\sec (315) = \dfrac{1}{{\cos ( - 45)}}$
Since we also know that the cosine is an even function, $cos( - x) = cos(x)$ , so
$\sec (315) = \dfrac{1}{{\cos (45)}}$
${45^ \circ }$ is a special angle and we know that the cosine of ${45^ \circ }$ is $\dfrac{1}{{\sqrt 2 }}$ ,
Therefore, we will get the required result ,
$\sec (315) = \dfrac{1}{{\dfrac{1}{{\sqrt 2 }}}} = \sqrt 2$
Hence we get the solution.
So, the correct answer is “$\sqrt 2$”.

Note : By using the basic trigonometric identity given below we can simplify and can find the value of the above expression that is $sec\left( {315} \right)$ . In order to solve and simplify the given expression we
have to use the identities and express our given expression in the simplest form and thereby solve
it. Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.