Question

How do you evaluate $\arccos \left( -\dfrac{1}{\sqrt{2}} \right)$?

Answer 1

Hint: To solve the inverse trigonometric identity $\arccos \left( -\dfrac{1}{\sqrt{2}} \right)$ then we need to remove the negative sign from the identity first of all. We have to apply the formula \[\arccos \left( -x \right)=\arccos x\]. This is done because \[\arccos \left( -x \right)\] comes in the fourth quadrant and cosine is always positive in the first and the fourth quadrant. The inverse trigonometric identity \[\arccos \left( x \right)\] could also be written as \[{{\cos }^{-1}}x\].

Complete step by step solution:
We have our given identity that is $\arccos \left( -\dfrac{1}{\sqrt{2}} \right).....\left( 1 \right)$.
We have to simplify the angle of the identity (1); we know that \[\arccos \left( -x \right)=\arccos x\] so, we have to write the identity $\arccos \left( -\dfrac{1}{\sqrt{2}} \right)$ in another form such that, we get:
\[\begin{align}
  & \Rightarrow \arccos \left( -\dfrac{1}{\sqrt{2}} \right) \\
 & \Rightarrow \arccos \left( \dfrac{1}{\sqrt{2}} \right).....\left( 2 \right) \\
\end{align}\]
Now, we have the identity (2). The identities are converted in simplified form because it will be easier to solve. Here we should use the table of angles and their identity. From the table we know that the value for $\arccos \left( \dfrac{1}{\sqrt{2}} \right)$ is \[\dfrac{\pi }{4}\] or \[45{}^\circ \] in degrees.
$\begin{align}
  & \arccos \left( \dfrac{1}{\sqrt{2}} \right) \\
 & \Rightarrow \dfrac{\pi }{4}.....\left( 3 \right) \\
\end{align}$

Now, we have obtained the solution to the problem in identity (3). The solution of the given identity is $\dfrac{\pi }{4}$.

Note:While solving an inverse trigonometric problem, we should always keep in mind that we must have learned all the formulas before solving the question, all the formulas are extremely necessary for solving the questions. We must 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. \[\dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{\pi }{3},\dfrac{\pi }{2}\]. Degrees and radians are 2 most important notations of angles.