Question

How do you evaluate \[\sin \left( {\dfrac{{7\pi }}{6}} \right)\] ?

Answer 1

Hint: For solving this question we will use the trigonometric identity and for using it we have to first change the degree at written in the form of ${180^ \circ } + \theta $ and as we know that $\sin \left( {\pi + \theta } \right) = - \sin \theta $ . So by substituting the values, and solving them we will be able to get the answer.

Formula used:
The trigonometric identity used is,
$\sin \left( {\pi + \theta } \right) = - \sin \theta $
Here, $\theta $ will be the angle

Complete step by step solution:
So we have to find the value of \[\sin \left( {\dfrac{{7\pi }}{6}} \right)\] and for this first of all we will change degree in such a way that the values will remain the same.
Therefore we can write it in form as
\[ \Rightarrow \sin \left( {\dfrac{{7\pi }}{6}} \right) = \sin \left( {\pi + \dfrac{\pi }{6}} \right)\]
And from the formula, we already know that $\sin \left( {\pi + \theta } \right) = - \sin \theta $
Therefore we can write the above expression as
\[ \Rightarrow \sin \left( {\dfrac{{7\pi }}{6}} \right) = - \sin \left( {\dfrac{\pi }{6}} \right)\]
And as we know the value of $\sin {30^ \circ } = \dfrac{1}{2}$ , therefore on substituting the values, the expression will become
\[ \Rightarrow \sin \left( {\dfrac{{7\pi }}{6}} \right) = - \dfrac{1}{2}\]
Hence, on evaluating \[\sin \left( {\dfrac{{7\pi }}{6}} \right)\] we get $ - \dfrac{1}{2}$ .

Note: This question will always be solved by using the identities, the only need in this type of question is our approach to think about how we can take the equation into any identity. So by practice, it will come. Also while solving since, degrees are not written directly and they are written in pie form. So we should be aware of that by not making mistakes while solving the question.