Question

TIf 1 degree=0.017 radian, then the approximate value of $ \sin ({{46}^{\circ }}) $ will be
a.0.7194
b. $ \dfrac{0.017}{\sqrt{2}} $
c. $ \dfrac{1.017}{\sqrt{2}} $
d.None of these

Answer 1

Hint:In this question, we are asked to find $ \sin ({{46}^{\circ }}) $ and are given the value of 1 degree in terms of radians. As the value of $ \sin ({{45}^{\circ }})=\dfrac{1}{\sqrt{2}} $ , we can express $ \sin ({{46}^{\circ }}) $ in terms of $ \sin ({{45}^{\circ }}) $ by using the formula for sine of a sum of two angles. Thereafter, as the value of $ {{1}^{\circ }} $ is very small, we can use the small angle approximation of sine and cosine functions, that is for $ x\approx 0 $ , $ \sin (x)\approx x $ and $ \cos (x)\approx 1 $ where x is in radians and use the values from it in the equations to obtain our desired answer.

Complete step-by-step answer:
We know that the formula for the sine of a sum of angles is given by
 $ \sin (a+b)=\sin a\cos b+\cos a\sin b................(1.1) $
Therefore, taking $ a={{45}^{\circ }} $ and $ b={{1}^{\circ }} $ in equation (1.1), we obtain
 $ \sin \left( {{46}^{\circ }} \right)=\sin \left( {{45}^{\circ }} \right)\cos \left( {{1}^{\circ }} \right)+\cos \left( {{1}^{\circ }} \right)\sin \left( {{45}^{\circ }} \right)...............(1.2) $
However, we know that the value of $ \sin \left( {{45}^{\circ }} \right)=\cos \left( {{45}^{\circ }} \right)=\dfrac{1}{\sqrt{2}} $ . Therefore, using these values in equation (1.2), we get
 $ \begin{align}
  & \sin \left( {{46}^{\circ }} \right)=\sin \left( {{45}^{\circ }} \right)\cos \left( {{1}^{\circ }} \right)+\cos \left( {{45}^{\circ }} \right)\sin \left( {{1}^{\circ }} \right) \\
 & =\dfrac{1}{\sqrt{2}}\times \cos \left( {{1}^{\circ }} \right)+\dfrac{1}{\sqrt{2}}\times \sin \left( {{1}^{\circ }} \right) \\
 & =\dfrac{1}{\sqrt{2}}\left( \cos \left( {{1}^{\circ }} \right)+\sin \left( {{1}^{\circ }} \right) \right).........................(1.3) \\
\end{align} $
Now, we note that as $ {{1}^{\circ }}\ll {{45}^{\circ }} $ and as $ {{1}^{\circ }} $ is very close to $ {{0}^{\circ }} $ , we can use the small angle approximation of sine and cosine which states that for \[x\approx 0\] ,
 $ \begin{align}
  & \sin x\approx x \\
 & \cos x\approx \cos \left( {{0}^{\circ }} \right)\approx 1...........................(1.4) \\
\end{align} $
Where x is given in radians. Therefore, taking the approximation in (1.4) with $ x={{1}^{\circ }} $ , as $ {{1}^{\circ }} $ is equivalent to 0.017 radians as given in the question, we get
 $ \begin{align}
  & \sin \left( {{1}^{\circ }} \right)\approx 0.017 \\
 & \cos \left( {{1}^{\circ }} \right)\approx 1............................(1.4a) \\
\end{align} $
 Using it in equation (1.3), we obtain
 $ \begin{align}
  & \sin \left( {{46}^{\circ }} \right)=\dfrac{1}{\sqrt{2}}\left( \cos \left( {{1}^{\circ }} \right)+\sin \left( {{1}^{\circ }} \right) \right) \\
 & \approx \dfrac{1}{\sqrt{2}}\left( 1+0.017 \right) \\
 & =\dfrac{1.017}{\sqrt{2}}.........................(1.5) \\
\end{align} $
Which matches option (c) given in the question. Therefore, the required answer is option (c).

Note: We should note that we should convert the value of x in radians before using it in equation (1.4), therefore, we cannot write $ \sin \left( {{1}^{\circ }} \right)\approx 1 $ because here the angle is in degrees and not in radians. Also, we should be careful to use the correct sign between the terms in equation (1.1), there should be a positive sign in case of sine of a sum of angles whereas the sign is negative if we expand the cosine of a sum of angles.