Question

How do you evaluate $\cot \left[ \arccos \left( \dfrac{-4}{5} \right) \right]$ ?

Answer 1

For answering this question we need to evaluate the value of $\cot \left[ \arccos \left( \dfrac{-4}{5} \right) \right]$. We need to assume that the value of $\arccos \left( \dfrac{-4}{5} \right)={{\cos }^{-1}}\left( \dfrac{-4}{5} \right)$is $x$ then we will have $\cos x=\dfrac{-4}{5}$ .
From the basic concepts we know that we have a formula in trigonometry given as $\sin x=\sqrt{1-{{\cos }^{2}}x}$ and $\cot x=\dfrac{\cos x}{\sin x}$ .

Complete step by step solution:
Now considering from the question we have been asked to evaluate the value of $\cot \left[ \arccos \left( \dfrac{-4}{5} \right) \right]$.
For answering this question we need to assume that the value of $\arccos \left( \dfrac{-4}{5} \right)={{\cos }^{-1}}\left( \dfrac{-4}{5} \right)$is $x$ then we will have $\cos x=\dfrac{-4}{5}$ .
From the basic concepts of trigonometry we know that we have formulae in trigonometry given as $\sin x=\sqrt{1-{{\cos }^{2}}x}$ and $\cot x=\dfrac{\cos x}{\sin x}$ .
Now we will derive the value of sine function from the assumed value of cosine function then we will have $\Rightarrow \sin x=\sqrt{1-{{\left( \dfrac{-4}{5} \right)}^{2}}}$ .
By simplifying this value we will have
$\begin{aligned} & \Rightarrow \sin x=\sqrt{1-\left( \dfrac{16}{25} \right)} \\\ & \Rightarrow \sin x=\sqrt{\dfrac{9}{25}} \\\ & \Rightarrow \sin x=\dfrac{3}{5} \\\ \end{aligned}$ .
Now we will derive the value of $\cot x$ then we will have
$\begin{aligned} & \Rightarrow \cot x=\dfrac{\cos x}{\sin x} \\\ & \Rightarrow \cot x=\left( \dfrac{\left( \dfrac{-4}{5} \right)}{\left( \dfrac{3}{5} \right)} \right) \\\ \end{aligned}$.
By simplifying we will have $\Rightarrow \cot x=\left( \dfrac{-4}{3} \right)$ .

Therefore we can conclude that the value of the given trigonometric expression $\cot \left[ \arccos \left( \dfrac{-4}{5} \right) \right]$ is $\dfrac{-4}{3}$

Note: While answering questions of this type we should be sure with our concepts if we have a good basic knowledge then we can solve this question in a short span of time and very few mistakes are possible. Similarly we can derive the value of any trigonometric expression for example if we consider a trigonometric expression given as $\tan \left( \arcsin \left( \dfrac{3}{5} \right) \right)$ we will have its value as $\Rightarrow \dfrac{3}{4}$ and can be derived in a similar form.