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# How do you evaluate $\sec \left( -\dfrac{\pi }{6} \right)$ ?

Last updated date: 21st Jul 2024
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Hint: We have to find the value of a negative angle of the secant function which is a trigonometric function. We shall use basic trigonometry to analyze the angle that has been referred to and then find its value in the secant function. For this, we must know the values of secant of the few principal angles in trigonometry derived by using the Pythagorean theory.

We understand that $-\dfrac{\pi }{6}$ lies in the fourth quadrant of the cartesian plane which consists of angles ranging from $\left( \dfrac{3\pi }{2},2\pi \right)$ in the anticlockwise sense and angles ranging from $\left( 0,-\dfrac{\pi }{2} \right)$ in the clockwise sense.
This implies that $\sec \left( -\dfrac{\pi }{6} \right)$ must have a positive value. Thus, it can be written as $\sec \left( \dfrac{\pi }{6} \right)$.
Now, by the basic values of trigonometric functions, we know that $\sec \left( \dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$.
$\Rightarrow \sec \left( -\dfrac{\pi }{6} \right)=\dfrac{2}{\sqrt{3}}$
Therefore, the value of $\sec \left( -\dfrac{\pi }{6} \right)$ is $\dfrac{2}{\sqrt{3}}$.