Question

Find the value of $\sec \dfrac{13{{\pi }^{c}}}{3}$.

Answer 1

In order to evaluate the value of $\sec \dfrac{13{{\pi }^{c}}}{3}$, we will be expressing $\dfrac{13{{\pi }^{c}}}{3}$ in such a way that we will be considering the principle angle to which $\dfrac{13{{\pi }^{c}}}{3}$ is nearer and then add the remaining angle measure so that it will be equal to the original angle given. And then we will be checking the quadrant it belongs to and decide if the value is positive or negative and then evaluate the value.

Complete step-by-step solution:
Let us have a brief regarding the trigonometric functions. The counter-clockwise angle between the initial arm and the terminal arm of an angle in standard position is called the principal angle. Its value is between ${{0}^{\circ }}$ and ${{360}^{\circ }}$. The relationship between the angles and sides of a triangle are given by the trigonometric functions. The basic trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant. These are the basic main trigonometric functions used.
Now let us find the value of $\sec \dfrac{13{{\pi }^{c}}}{3}$.
We can express $\sec \dfrac{13{{\pi }^{c}}}{3}$ as $\sec \dfrac{13\pi }{3}=\sec \left( 4\pi +\dfrac{\pi }{3} \right)$.
We get $\sec \dfrac{13\pi }{3}=\sec \dfrac{\pi }{3}$ as $4\pi +\theta$ will be in the first quadrant. Since it is in the first quadrant, $\sec \left( 4\pi +\theta \right)$ is positive.
$\therefore$ The value of $\sec \dfrac{13{{\pi }^{c}}}{3}$ is $2$.

Note: In order to convert from radians to degrees, we have to multiply the radians by ${{180}^{\circ }}\pi$ radians. We use radians in trigonometry because they make it possible to relate a linear measure and an angle measure. The common error could be not converting the degree to radians or radians to degrees with accurate values or simply assuming the wrong values.