Question

How do you find the exact value of $\arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)$?

Answer 1

In this problem, we have to find the exact value of the given trigonometric expression. We can find the value by using some trigonometric formula. We can take the values required for the formula from the given expression and substitute the values in the formula to find the exact value. We will use degree values in this problem to find the exact value.

Complete step by step answer:
We know that the given trigonometric expression is,
$\arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)$ …… (1)
We know that the trigonometric formula can be used in this problem is,
$\arctan x+\arctan y+\arctan z=\arctan \dfrac{x+y+z-xyz}{1-xy-yz-zx}$ …….. (2)
We can take the values from (1) to be substituted in the above formula,
x = $\dfrac{1}{2}$, y = $\dfrac{1}{5}$, z = $\dfrac{1}{8}$.
We can substitute the above value in the formula (2), we get
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\arctan \dfrac{\dfrac{1}{2}+\dfrac{1}{5}+\dfrac{1}{8}-\left( \dfrac{1}{2}\times \dfrac{1}{5}\times \dfrac{1}{8} \right)}{1-\left( \dfrac{1}{2}\times \dfrac{1}{5} \right)\left( \dfrac{1}{5}\times \dfrac{1}{8} \right)\left( \dfrac{1}{8}\times \dfrac{1}{2} \right)}$
Now we can simplify the above step, we get
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\arctan \dfrac{0.5+0.2+0.125-\left( 0.5\times 0.2\times 0.125 \right)}{1-\left( 0.5\times 0.2 \right)\left( 0.2\times 0.125 \right)\left( 0.125\times 0.5 \right)}$
Now we can add the number in the numerator and multiply the numbers in the denominator.
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\arctan \dfrac{0.8125}{0.8125}$
We can now cancel the similar terms, we get
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\arctan \left( 1 \right)$
We know that $\tan \dfrac{\pi }{4}=1$, substituting this value in the right-hand side of the equation, we get
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\arctan \left( \tan \dfrac{\pi }{4} \right)$
We can cancel the tangent and arctan, we get
$\Rightarrow \arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\dfrac{\pi }{4}$.
Therefore, the value of $\arctan \left( \dfrac{1}{2} \right)+\arctan \left( \dfrac{1}{5} \right)+\arctan \left( \dfrac{1}{8} \right)=\dfrac{\pi }{4}$.

Note:
Students make mistakes while writing the correct formula, which should be concentrated. We should know that to solve these types of problems, we have to know trigonometric formulas, identities, properties and degree values to get the required answer.