Question

How do you calculate $\left[ {{{\tan }^{ - 1}}\left( {\left( {\dfrac{1}{2}} \right)} \right)} \right]$.

Answer 1

In this question, we have a trigonometric inverse function. The trigonometric inverse function is also called the arc function. To solve the trigonometric inverse function we assume the angle $\theta$ which is equal to that trigonometric inverse function. Then we find the value of $\theta$.

Complete step by step solution:
In this question, we used the word trigonometric inverse function. The trigonometric inverse function is defined as the inverse function of trigonometric identities like sin, cos, tan, cosec, sec, and cot. The trigonometric inverse function is also called cyclomatic function, anti trigonometric function, and arc function. The trigonometric inverse function is used to find the angle of any trigonometric ratio. The trigonometric inverse function is applicable for right-angle triangles.
Let us discuss all six trigonometric functions.
Arcsine function: it is the inverse function of sine. It is denoted as ${\sin ^{ - 1}}$.
Arccosine function: it is the inverse function of cosine. It is denoted as ${\cos ^{ - 1}}$.
Arctangent function: it is the inverse function of tangent. It is denoted as ${\tan ^{ - 1}}$.
Arccotangent function: it is the inverse function of cotangent. It is denoted as ${\cot ^{ - 1}}$.
Arcsecant function: it is the inverse function of secant. It is denoted as ${\sec ^{ - 1}}$.
Arccosecant function: it is the inverse function of cosecant. It is denoted as $\cos e{c^{ - 1}}$.
Now, we come to the question. The data is given below.
$\left[ {{{\tan }^{ - 1}}\left( {\left( {\dfrac{1}{2}} \right)} \right)} \right]$
Let us assume that the angle $\theta$ (angle of the right-angle triangle) is equal to that trigonometric function.
Then,
$\Rightarrow \theta = \left[ {{{\tan }^{ - 1}}\left( {\left( {\dfrac{1}{2}} \right)} \right)} \right]$
Then,
$\Rightarrow \tan \theta = \dfrac{1}{2}$
We find the value of angle$\theta$.
Then,
$\Rightarrow \theta = \left[ {{{\tan }^{ - 1}}\left( {\left( {\dfrac{1}{2}} \right)} \right)} \right]$
After calculating the above, the result is as below.
$\therefore \theta = 26.57^\circ$

Therefore, the value of $\left[ {{{\tan }^{ - 1}}\left( {\left( {\dfrac{1}{2}} \right)} \right)} \right]$ is $26.57^\circ$.

Note:
If you have a trigonometric inverse function with value. Then first assume that the angle $\theta$. Then find the value of that angle $\theta$. The angle $\theta$ is the angle of the right-angle triangle. And trigonometric functions are always used for right-angle triangles.