Question

How do you calculate $\sec \left( \dfrac{13\pi }{4} \right)$

Answer 1

Now we will first try to write the angle such that it is in the range of $\left( 0,2\pi \right)$ . Using this we will use different properties of cos such as $\cos \left( 2\pi +\theta \right)=\cos \theta$ and $\cos \left( \pi +\theta \right)=-\cos \theta$ . Hence using this we will try to find the value of $\cos \left( \dfrac{15\pi }{4} \right)$ .

Complete step by step answer:
Now we are given the sec function.
We know that the sec is a trigonometric ratio which gives us the ratio of Hypotenuse and adjacent side.
Now we know that cos is nothing but the inverse function of sec.
We will now try to use different trigonometric properties to simplify the expression and find its value.
Now consider $\sec \left( \dfrac{13\pi }{4} \right)$
We can write $13\pi =8\pi +5\pi$ now using this we get the equation as $\sec \left( \dfrac{8\pi +5\pi }{4} \right)$
Now separating the terms in sec function we get,
$\Rightarrow \sec \left( \dfrac{8\pi }{4}+\dfrac{5\pi }{4} \right)$
$\Rightarrow \sec \left( 2\pi +\dfrac{5\pi }{4} \right)$
Now we know that $\sec \theta =\dfrac{1}{\cos \theta }$ . Hence we will try to find the value of $\cos \left( 2\pi +\dfrac{5\pi }{4} \right)$
Now we know that $\cos \left( 2\pi +\theta \right)=\cos \theta$ .
Hence we get,
$\Rightarrow \cos \left( 2\pi +\dfrac{5\pi }{4} \right)=\cos \left( \dfrac{5\pi }{4} \right)$
Now again we will write $5\pi =4\pi +\pi$, Hence we get the equation as,
$\Rightarrow \cos \left( 2\pi +\dfrac{5\pi }{4} \right)=\cos \left( \dfrac{4\pi }{4}+\dfrac{\pi }{4} \right)$
$\Rightarrow \cos \left( 2\pi +\dfrac{5\pi }{4} \right)=\cos \left( \pi +\dfrac{\pi }{4} \right)$
Now we know that $\cos \left( \pi +\theta \right)=-\cos \theta$ hence using this we get,
$\Rightarrow \cos \left( 2\pi +\dfrac{5\pi }{4} \right)=-\cos \left( \dfrac{\pi }{4} \right)$
Now we know that the value of $\cos \left( \dfrac{\pi }{4} \right)$ is equal to $\dfrac{1}{\sqrt{2}}$
Hence using this we get
$\Rightarrow \cos \left( 2\pi +\dfrac{5\pi }{4} \right)=-\dfrac{1}{\sqrt{2}}$
Now using invertendo property we get the equation as,
$\begin{aligned} & \Rightarrow \dfrac{1}{\cos \left( 2\pi +\dfrac{5\pi }{4} \right)}=-\sqrt{2} \\\ & \Rightarrow \sec \left( 2\pi +\dfrac{5\pi }{4} \right)=-\sqrt{2} \\\ \end{aligned}$
Hence we have the value of $\sec \left( \dfrac{13\pi }{4} \right)=-\sqrt{2}$

Note:
Now note that while calculating we can directly calculate the value of sec after simplifying. We can also use the property $\sec \left( \theta -2\pi \right)=\sec \theta$ and write the expression in this form to find the value of $\sec \left( \dfrac{3\pi }{4} \right)$ . Hence we can easily find the value of the expression.