Question

If $\tan \theta = \dfrac{2}{3}$, find $\theta $?

Answer 1

Hint: The ratio between the perpendicular of the right-angled triangle and the base of the right-angled triangle is called the tangent function. A right- angled triangle is the one that has one angle equal to $90^\circ $. In this question we have to find the value of $\theta $ . It can be calculated by taking the inverse of tangent function.

Complete answer:
In this question we have to find the value of $\theta $ . It can be calculated by taking the inverse of the tangent function.
Inverse tangent is one of the trigonometric functions. Each trigonometric function has an inverse of it, whether it is sine, cosine, tangent, secant, cosecant and cotangent. These functions are also widely used, apart from the trigonometric formulas, to solve many problems. Inverse functions are also called Arc functions because they give the length of the arc for a given value of trigonometric functions.
The inverse tangent function is the inverse of the tangent function and is used to obtain the values of angles for a right-angled triangle.
Let $\tan A = 1$
Then, $A = {\tan ^{ - 1}}1$
The value of $\tan $ and ${\tan ^{ - 1}}$ is the same.
So $A = 40^\circ $ or $A = \dfrac{\pi }{4}$
In the question we have given that $\tan \theta = \dfrac{2}{3}$ so we have to find the value of $\theta $
Therefore, $\theta = {\tan ^{ - 1}}\left( {\dfrac{2}{3}} \right)$
$ \Rightarrow \theta \approx 0.5880$
Hence, the value of $\theta\approx 0.5880$ (in radians)

Note: The value of $\tan \theta $ lies between $\left[ {0,\dfrac{\pi }{2}} \right]$ therefore the value of $\theta $ is also lies between $\left[ {0,\dfrac{\pi }{2}} \right]$.
$\dfrac{2}{3}$ is not one of the special values of the basic trigonometric functions. So, it is difficult to express $\theta $ exactly in this case. Some of the so-called “special angles” and their corresponding outputs for the tangent functions are: $\theta = 0 \Rightarrow \tan \theta = 0$, $\theta = \dfrac{\pi }{4} \Rightarrow \tan \theta = 1$,$\theta = \dfrac{\pi }{3} \Rightarrow \tan \theta = \sqrt 3 $.