Question

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

Answer 1

Hint:
We are given an inverse trigonometric function, which gives us an angle after solving. The given expression $\arctan \left( {\dfrac{1}{2}} \right)$ represents an angle for which the ratio of perpendicular and base is $\dfrac{1}{2}$ . Use an inverse tangent data table to find the exact value.

Complete step by step solution:
Here in this question, we need to find the exact value of the expression $\arctan \left( {\dfrac{1}{2}} \right)$ .
Before starting with the solution of the problem we must understand the concept of the inverse tangent function. The arctangent or inverse tangent function is the inverse of the tangent function. It returns the angle whose tangent is a given number. For every trigonometry function, there is an inverse function that works in reverse. These inverse functions have the same name but with 'arc' in front. So the inverse of tan is arctan etc. When we see "arctan x", we understand it as "the angle whose tangent is x"
The range of inverse tangent function will be $\left( { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right)$ or $\left( {90^\circ ,90^\circ } \right)$ and the domain is the set of all real numbers.
${\text{So}},{\text{ }}\tan 30^\circ = 0.577$ , means the tangent of $30^\circ $ is $0.577$
Then, $\arctan \left( {0.577} \right) = 30^\circ $ , means the angle whose tangent is $0.577$ is $30^\circ $
Similarly, for the given case of $\arctan \left( {\dfrac{1}{2}} \right)$ , we need to find the angle for which the tangent function will attain the value $\dfrac{1}{2}$ .
Remember that this value $\dfrac{1}{2}$ represents the ratio of perpendicular and base of a right-angled triangle having one of its non-right angles as $\arctan \left( {\dfrac{1}{2}} \right)$ .
Since, there is a way to find the exact value of the inverse tangent function, other than using a calculator or using a data table which consists of values of all the inverse trigonometric functions.

Using the inverse tangent function data table, we get:
$ \Rightarrow \arctan \left( {\dfrac{1}{2}} \right) = 26.6^\circ $

Note:
Remember that the fundamental definition of tangent function states that it represents the ratio of the perpendicular side and base side of a right-angled triangle. According to this definition, the given expression $\arctan \left( {\dfrac{1}{2}} \right)$ represents an angle for which the ratio of perpendicular and base is $\dfrac{1}{2}$ . Since there is no commonly known value of tangent that gives us $\dfrac{1}{2}$ or $0.5$ , we can draw a right-angled triangle with perpendicular of length $1unit$ and base of length $2units$ . After completing the triangle, you can measure the smallest angle formed in the triangle. This will give us the required answer.