Question

In the adjoining figure, find the value of $\angle ABC=?$

Answer 1

Hint: The given triangle has lengths of two mutually perpendicular sides and we need to find the angle value, we can apply trigonometric ratio over here. As tangent is the ratio of perpendicular to base dimension and gives the relationship with corresponding angle it can be used.

Complete step by step answer:
The longest side of the right-angle triangle is known as Hypotenuse(H) while the side opposite to the given angle, say$(\theta )$, is known as perpendicular (P) and the third one adjacent to angle is known as base (B). The three basic trigonometric functions are sine, cosine and tangent for a given angle. The functions are calculated as, $\sin \theta =\dfrac{P}{H},\cos \theta =\dfrac{B}{H},\tan \theta =\dfrac{P}{B}$. The ratios for angle can be calculated by equating the ratio to standard function value as per table below:
$\begin{matrix}
   function & 0{}^\circ & 30{}^\circ & 45{}^\circ & 60{}^\circ & 90{}^\circ \\
   \sin & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1 \\
   \cos & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} & 0 \\
   \tan & 0 & \dfrac{1}{\sqrt{3}} & 1 & \sqrt{3} & \infty \\
\end{matrix}$
We have been given the following sides and right angle at vertex A:
$AC=100\sqrt{3}$
$AB=100$
Looking to the given figure we can see that the angle under consideration is $\angle ABC$ and the given sides of the right angle triangle are perpendicular (AC) and base (AB) to the $\angle ABC$. Using the tangent function, we can calculate the angle.
$\tan \theta =\dfrac{P}{B}$
Substituting the corresponding values, we get
$\Rightarrow \tan \angle ABC=\dfrac{AC}{AB}$
Substituting the values of AB and AC as given, we get,
$\Rightarrow \tan \angle ABC=\dfrac{100\sqrt{3}}{100}$
$\Rightarrow \tan \angle ABC=\sqrt{3}.................(i)$
We know that, $\tan {{60}^{{}^\circ }}$ is $\sqrt{3}$.
Substituting in equation (i), we get
$\tan \angle ABC=\tan {{60}^{{}^\circ }}$
$\Rightarrow \angle ABC={{60}^{{}^\circ }}$

Hence, the final answer is ${{60}^{{}^\circ }}$.

Note: The chances of doing mistakes in substituting the value of $\tan {{60}^{{}^\circ }}$ as it is little similar to $\tan {{30}^{{}^\circ }}=\dfrac{1}{\sqrt{3}}$ and there may be confusion while substituting. In this case we will get the wrong answer.
Here you can use other ratios as well.