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In figure, calculate the value of $\theta$$A){\text{ }}36.9_{}^0$$B){\text{ }}38.6_{}^0$$C){\text{ }}41.4_{}^0$$D){\text{ }}47.2_{}^0$$E){\text{ }}51.3_{}^0$

Last updated date: 17th Jun 2024
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Hint: To solve this question we can apply the formula, of $\tan \theta$, and then we will get the required value of $\theta$ after doing inverse at last to get the required answer.

Formula used:
$\tan \theta = \dfrac{{Perpendicular}}{{Base}}$.

Complete step by step answer:
From the above diagram, we can see that the value of the perpendicular is $5$ and the value of the base is $4$.
Now by applying the formula $\tan \theta = \dfrac{{Perpendicular}}{{Base}}$ we get-
$\tan \theta = \dfrac{5}{4}$
$\Rightarrow \theta = {\tan ^{ - 1}}\left( {\dfrac{5}{4}} \right)$
$\Rightarrow \theta = 51.3_{}^0$

$\therefore$ The required value of $\theta$ is $51.3_{}^0$. Thus the correct option is $E$.

Note:
It is to be kept in mind that there are several formulas of $\tan \theta$ and most of the students make mistakes in applying the correct formula so try to remember all the formulas.
Another formula of $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$or $\tan \theta = \sqrt {\sec _{}^2\theta - 1}$
There are three sides in a triangle and each side represents either perpendicular or base or hypotenuse. So when the values of perpendicular and hypotenuse are given, then we can find out the value of $\theta$ by using the formula$\sin \theta = \dfrac{{Perpendicular}}{{Hypotenuse}}$. Other formulas for finding out the values of $\theta$ is $\sin \theta = \sqrt {1 - \cos _{}^2\theta }$
When the values of base and hypotenuse are given then we can find out the value of $\theta$by using the formula of $\cos \theta$$= \dfrac{{Base}}{{Hypotenuse}}$. Other formula for finding out the value of $\cos \theta$ is $\cos \theta = \sqrt {1 - \sin _{}^2\theta }$
When the values of perpendicular and base are given, then we can find out the value of $\theta$ by using the formula of $\tan \theta = \dfrac{{Perpendicular}}{{Base}}$.
If the values of perpendicular, base, and height or any two of them are given in the question, then to find out the angle we will apply the formula of $\sin \theta$, $\cos \theta$ or $\tan \theta$