Question

How do you evaluate sec ($ - \pi $)?

Answer 1

Hint: sec x is reciprocal of cos x. The range of cos x is $ - 1 \leqslant \cos x \leqslant 1$ . So as sec x is reciprocal of cos x, therefore, sec x = $\dfrac{1}{{\cos x}}$ . Now, note that when the numerator is base and denominator is hypotenuse is known as cos x that means cos x = $\dfrac{{Base}}{{Hypotenuse}}$ . So as secant is reciprocal of cos x that means when the numerator is hypotenuse and numerator is the base is known as sec x as sec x = $\dfrac{1}{{\cos x}} = \dfrac{1}{{\dfrac{{Base}}{{Hypotenuse}}}} = \dfrac{{Hypotenuse}}{{Base}}$ . Therefore, sec x = $\dfrac{{Hypotenuse}}{{Base}}$ .

Complete step by step solution:
We have to find the value of sec (- $\pi $ ). As we know cos x = $\dfrac{1}{{\sec x}}$.
 $\therefore \sec ( - \pi ) = \dfrac{1}{{\cos ( - \pi )}}$ ---- (i)
We know that
 $ \Rightarrow \cos (\pi ) = - 1$
 $\therefore \cos ( - \pi ) = - ( - 1) = 1$
Then we just need to remember that the value of cos($ - \pi $) = 1 [where $\pi $in degrees is ${180^ \circ }$]
Putting the value of cos($ - \pi $) in equation (i)
$ \Rightarrow \sec ( - \pi ) = \dfrac{1}{1} = 1$
$\therefore \sec ( - \pi ) = 1$

So, the value of $\sec ( - \pi )$will be -1.

Additional Information:
We can get the value of cos x by dividing the base with hypotenuse. And cos x is the reciprocal of sec x that is $\cos x = \dfrac{1}{{\sec x}}$. So, we can find the value of sec x by dividing the hypotenuse with base in a right-angle triangle.

Note:
We can find sec x when the length of the hypotenuse is divided with the base, it gives sec x in a right-angle triangle. And cos x is reciprocal of sec x so we can also find the value of cos x with the help of sec x.