Question

How do you evaluate $\cos \dfrac{\pi }{6}+\sin \dfrac{\pi }{6}$?

Answer 1

Hint: If we want to solve the trigonometric identity $\cos \dfrac{\pi }{6}+\sin \dfrac{\pi }{6}$ then remember the table of the angles and their identities. Put the value of the identities of their angles. First of all convert the identity into all cosine identity.

Complete step by step solution:
We have our given identity that is $\cos \dfrac{\pi }{6}+\sin \dfrac{\pi }{6}.....\left( 1 \right)$.
The \[\sin \dfrac{\pi }{6}\] can be written as \[\cos \left( \dfrac{\pi }{2}-\dfrac{\pi }{6} \right)\] so, we have to write the identity into all cosine form such that, we get:
$\begin{align}
  & \Rightarrow \cos \dfrac{\pi }{6}+\sin \dfrac{\pi }{6} \\
 & \Rightarrow \cos \dfrac{\pi }{6}+\cos \left( \dfrac{\pi }{2}-\dfrac{\pi }{3} \right).....\left( 2 \right) \\
\end{align}$
Now, we have the identity (2). The identities are converted in cosine form because it will be easier to solve.
$\begin{align}
  & \Rightarrow \cos \dfrac{\pi }{6}+\cos \left( \dfrac{\pi }{2}-\dfrac{\pi }{3} \right) \\
 & \Rightarrow \cos \dfrac{\pi }{6}+\cos \dfrac{\pi }{3}.....\left( 3 \right) \\
\end{align}$
Now, we have obtained the identity (3). We know that \[\cos \dfrac{\pi }{6}\] is equal to \[\dfrac{\sqrt{3}}{2}\]. So apply it in the identity.
$\begin{align}
  & \Rightarrow \cos \dfrac{\pi }{6}+\cos \dfrac{\pi }{3} \\
 & \Rightarrow \dfrac{\sqrt{3}}{2}+\cos \dfrac{\pi }{3}.....\left( 4 \right) \\
\end{align}$
Now, we have obtained the identity (4). We also know that \[\cos \dfrac{\pi }{3}\]is equal to \[\dfrac{1}{2}\]. So apply it in the identity.
$\begin{align}
  & \Rightarrow \dfrac{\sqrt{3}}{2}+\cos \dfrac{\pi }{3} \\
 & \Rightarrow \dfrac{\sqrt{3}}{2}+\dfrac{1}{2} \\
\end{align}$
Since we have got the numeric form as fraction, add the fractions to get the value of the identity. So adding both of the values, we get:
$\begin{align}
  & \Rightarrow \dfrac{\sqrt{3}}{2}+\dfrac{1}{2} \\
 & \Rightarrow \dfrac{\sqrt{3}+1}{2} \\
\end{align}$
Now, we have obtained the solution to the problem. The solution of the given identity is $\dfrac{\sqrt{3}+1}{2}$.

Note: We should always keep in mind that while solving a trigonometric problem, we should have learned all the formulas before solving the question. To solve trigonometry the main requirement is the formulas. Always remember the value of different angles of the identity, especially of the angles \[30{}^\circ ,45{}^\circ ,60{}^\circ ,90{}^\circ \] or we can use the radian angles i.e. \[\dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{\pi }{3},\dfrac{\pi }{2}\].