Question

How do you calculate $\arctan (1)?$

Answer 1

Hint: Arcus tangent is an inverse function of the tangent. Here we will assume the given expression as any angle such as theta and then by using the trigonometric identity and also by using All STC rule then will find the required value.

Complete step-by-step answer:
Take the given expression-
$\arctan (1)$
Let us assume that the given expression is equal to angle $\theta $. So it can be expressed as-
$\arctan (1) = \theta $
Multiply both the sides of the equation with tangent.
$\tan (\arctan (1)) = \tan \theta $
Since arc tangent is the inverse function, so arc tangent and tangent cancel each other.
$ \Rightarrow 1 = \tan \theta $
By referring to different angles for tangent we found that $\tan 45^\circ = 1$.
Since tangent is positive in the first quadrant by the law of All STC rule.
$ \Rightarrow \tan 45^\circ = \tan \theta $
By comparing both the sides of the equation –
$ \Rightarrow \theta = 45^\circ $
The above expression can be re-written as –
$ \Rightarrow \theta = \dfrac{\pi }{4}$
This is the required solution.

Additional Information:
Also remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).

Note: Arctan can also be expressed as tan inverse and can be denoted as ${\tan ^{ - 1}}$ . Remember the trigonometric table to substitute the values of different trigonometric function angles for an accurate and an efficient solution. Also, remember different trigonometric identities.