Question

How do you evaluate $\tan \left( {\arctan \left( {0.88} \right)} \right)$?

Answer 1

We are given a trigonometric expression. We have to find the value of the expression. First, evaluate the expression inside the brackets. Then, apply the trigonometric property to simplify the expression.

Complete step by step solution:
The given trigonometric expression is $\tan \left( {\arctan \left( {0.88} \right)} \right)$

It can be written as $\tan \left( {ta{n^{ - 1}}\left( {0.88} \right)} \right)$

The inverse of the function is opposite of the trigonometric function. Here, tan and $ta{n^{ - 1}}$ are opposite to each other.

When the function and its inverse is multiplied, then the result of multiplication is 1.

$\Rightarrow \tan \left( {ta{n^{ - 1}}\left( {0.88} \right)} \right) = 0.88$

Hence the value of $\tan \left( {\arctan \left( {0.88} \right)} \right)$ is equal to $0.88$.

Note: The students must note that we can also find the value of the given expression using the inverse tangent function data table.
First, the value of $ta{n^{ - 1}}\left( {0.88} \right)$ is determined from the table.
$\Rightarrow ta{n^{ - 1}}\left( {0.88} \right) = 41.35^\circ$

Now, we will substitute $41.35^\circ$ into the expression, we get:

$\Rightarrow \tan \left( {41.35^\circ } \right)$

Then, we will determine the value of $\tan \left( {41.35^\circ } \right)$ from the data table.

$\Rightarrow \tan \left( {41.35^\circ } \right) = 0.88$

Therefore, $\tan \left( {ta{n^{ - 1}}\left( {0.88} \right)} \right) = 0.88$