Question

Which is greater, \[\tan 1\] or \[{\tan ^{ - 1}}\left( 1 \right)\]?

Answer 1

Hint:
Here, we need to find which of the two given trigonometric ratios is greater. We will use the fact that the number \[\dfrac{\pi }{4}\] is less than the number 1. We will form an inequation using this fact. Then, we will form two inequalities, showing the relation between 1, and the two given numbers. Finally, we will observe the two given inequations to find which of the two given ratios is greater.

Formula Used:
The tangent of an angle \[\tan \theta = x\] can be written as a trigonometric inverse ratio as \[{\tan ^{ - 1}}\left( x \right) = \theta \].

Complete step by step solution:
We can write the number 1 as the fraction \[\dfrac{{28}}{{28}}\].
The number \[\dfrac{{28}}{{28}}\] is greater than the number \[\dfrac{{22}}{{28}}\].
We can write this as the inequation
\[ \Rightarrow \dfrac{{28}}{{28}} > \dfrac{{22}}{{28}}\]
Rewriting \[\dfrac{{28}}{{28}}\] as 1, we get
\[ \Rightarrow 1 > \dfrac{{22}}{{28}}\]
Rewriting 28 as the product of 4 and 7, we get
\[\begin{array}{l} \Rightarrow 1 > \dfrac{{22}}{{7 \times 4}}\\ \Rightarrow 1 > \dfrac{{22}}{7} \times \dfrac{1}{4}\end{array}\]
Substituting \[\dfrac{{22}}{7} = \pi \] in the inequation, we get
\[\begin{array}{l} \Rightarrow 1 > \pi \times \dfrac{1}{4}\\ \Rightarrow 1 > \dfrac{\pi }{4} \ldots \ldots \ldots \left( 1 \right)\end{array}\]
Now, we know that the value of tangent of an angle measuring \[\dfrac{\pi }{4}\] is equal to 1.
Thus, we get
\[\tan \dfrac{\pi }{4} = 1\]
The tangent of an angle \[\tan \theta = x\] can be written as a trigonometric inverse ratio as \[{\tan ^{ - 1}}\left( x \right) = \theta \].
Rewriting the equation, we get
\[{\tan ^{ - 1}}\left( 1 \right) = \dfrac{\pi }{4}\]
Substituting \[\dfrac{\pi }{4} = {\tan ^{ - 1}}\left( 1 \right)\] in inequation \[\left( 1 \right)\], we get
\[ \Rightarrow 1 > {\tan ^{ - 1}}\left( 1 \right) \ldots \ldots \ldots \left( 2 \right)\]
Taking the tangent of both sides of inequation \[\left( 1 \right)\], we get
\[ \Rightarrow \tan 1 > \tan \dfrac{\pi }{4}\]
Substituting \[\tan \dfrac{\pi }{4} = 1\] in the inequation, we get
\[ \Rightarrow \tan 1 > 1 \ldots \ldots \ldots \left( 3 \right)\]
Now, from inequations \[\left( 2 \right)\] and \[\left( 3 \right)\], we can observe that
\[ \Rightarrow \tan 1 > 1 > {\tan ^{ - 1}}\left( 1 \right)\]
Therefore, we get
\[ \Rightarrow \tan 1 > {\tan ^{ - 1}}\left( 1 \right)\]

Thus, we can conclude that \[\tan 1\] is greater than \[{\tan ^{ - 1}}\left( 1 \right)\].

Note:
Here we have taken the value of \[\dfrac{\pi }{4}\] as \[\dfrac{{22}}{{28}}\] because \[\pi \] is equal to \[\dfrac{{22}}{7}\]. We have taken 1 as \[\dfrac{{28}}{{28}}\] because we required the fractions with equal denominators as it makes the comparison and calculation a lot easier. The number \[\dfrac{{28}}{{28}}\] is greater than the number \[\dfrac{{22}}{{28}}\]. This is because if the denominators of two fractions are the same, then the fraction with the greater numerator is the greater fraction.