Question

How do you find the value of \[sec\left( 7\pi /6 \right)\]?

Answer 1

Hint: secant function, shortly called as sec is a trigonometric function which is calculated using the value of cos function since sec function is the reverse of the cosine function that is \[secx=\dfrac{1}{\cos x}\]. The domain of sec function is the set of all real numbers except those where the value of cos function is zero that is because at the points the value of secant function will be undefined. These values include \[n\pi +\dfrac{\pi }{2}\]. And the range of secant functions is \[\left( -\infty ,-1 \right)\] and \[\left( 1,\infty \right)\].

Complete step by step solution:
We have to find the value of \[sec\left( 7\pi /6 \right)\]
For this , we will first see that \[\dfrac{7\pi }{6}\] lies in the domain of the function
\[\dfrac{7\pi }{6}=\pi +\dfrac{\pi }{6}\], which is not equal to \[n\pi +\dfrac{\pi }{2}\] for any n.
So, \[\dfrac{7\pi }{6}\] lies in the domain of the function and the function is well defined.
Now , we can write \[sec\left( 7\pi /6 \right)\] as
\[sec\left( 7\pi /6 \right)=\sec (\pi +{}^{\pi }/{}_{6})\]
Now from the formula of \[sec\], we have
\[secx=\dfrac{1}{\cos x}\]
So,
\[sec\left( 7\pi /6 \right)=\sec (\pi +{}^{\pi }/{}_{6})=\dfrac{1}{\cos (\pi +{}^{\pi }/{}_{6})}\]
Now , we will use the trigonometric identity for \[cos\left( A+B \right)\]
\[~cos\left( A+B \right)=cosA.cosB-sinA.sinB\]
 Here , we can take \[A=\pi \] and \[B=\pi /6\]
So, we get \[cos\left( \pi +\dfrac{\pi }{6} \right)=cos\pi .cos\dfrac{\pi }{6}-sin\pi .sin\dfrac{\pi }{6}\]
Now, we will use the value of \[cos\pi \text{ },cos\pi /6\text{ },sin\pi \] and \[sin\pi /6\].
Since
\[\]\[\begin{array}{*{35}{l}}
   cos\pi =-1 \\
   cos\dfrac{\pi }{6}=\surd 3/2 \\
   sin\pi =0 \\
   sin\dfrac{\pi }{6}={\scriptscriptstyle 1\!/\!{ }_2} \\
\end{array}\]
Using these values in the above equation, we get
\[\begin{array}{*{35}{l}}
   cos\left( 7\pi /6 \right)=cos\left( \pi +\pi /6 \right)=cos\pi .cos\pi /6-sin\pi .sin\pi /6=\left( -1 \right)\left( \surd 3/2 \right)-0.\left( {\scriptscriptstyle 1\!/\!{ }_2} \right) \\
   =-\surd 3/2 \\
\end{array}\]

Thus we get the value of \[sec\left( 7\pi /6 \right)\]as \[-\dfrac{\sqrt{3}}{2}\].

Note:
As we have a reverse of cosine function which is called secant. Similarly, we have reciprocals of sine as well as tangent function, namely cosecant and cotangent functions respectively. So, we can use the same method to find the value of these reciprocals.