Question

Find $ \theta $ if $ \sin \theta =0.5798 $ .
(a) $ 57{}^\circ 2{6}' $
(b) $ 45{}^\circ 2{6}' $
(c) $ 35{}^\circ 2{6}' $
(d) None of these

Answer 1

Hint: We will look at the definition of the sine function. Then we will see the definition of an inverse trigonometric function. The value for the angle will be calculated by taking the inverse of the given sine function. We will see the importance of the concept of the principal value of inverse trigonometric functions. We will use the fact $ 1{}^\circ =6{0}' $ to convert the angle from decimal degrees to degrees and minutes.

Complete step by step answer:
In a triangle, for an angle $ \theta $ , the sine function is defined as $ \sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}} $ . If we are given the value of the sine function, it is possible to find the angle at which we obtain that value for the sine function. We can do this by taking the inverse of the trigonometric function. The inverse trigonometric functions give us the value of the angle by using any one of the trigonometry ratios. The inverse trigonometric function is a multi-valued function. Therefore, a range is defined for every inverse trigonometric function. It is possible to determine this range since the values obtained repeat for every function in some interval. So, the value that lies specifically in the determined range is called as the principal value of the inverse trigonometric function. We can obtain the value of $ \theta $ by taking the inverse sine function. We are given that $ \sin \theta =0.5798 $ . Therefore, we have the following,
 $ \theta ={{\sin }^{-1}}\left( 0.5798 \right) $
This value obtained is the following,
 $ \theta =35.44{}^\circ $
Now, we know that $ 1{}^\circ =6{0}' $ . We will use this fact to convert the angle from decimal degrees to degrees and minutes. We have,
 $ \begin{align}
  & \theta =35.44{}^\circ \\
 & \therefore \theta =35{}^\circ +0.44{}^\circ \\
\end{align} $
Therefore, using the conversion, we get
 $ \begin{align}
  & 0.44{}^\circ ={{\left( 0.44\times 60 \right)}^{\prime }} \\
 & \therefore 0.44{}^\circ ={{26.4}^{\prime }} \\
\end{align} $

Hence, $ \theta =35{}^\circ 2{6}' $ . Therefore, the correct option is (c).

Note:
If we know the value of $ \sin \theta $ for $ \theta =30{}^\circ ,45{}^\circ ,60{}^\circ $ in decimal form, then we can eliminate options (a) and (b) since the value of sine function for those angles will be close to the value of $ \sin \theta $ for $ \theta =45{}^\circ ,60{}^\circ $ . We just have to check if the given value of the sine function is equal to the value of sine function for the angle $ \theta =35{}^\circ 2{6}' $ and then select the correct option.