Question

Using differentials find the approximate value of $\tan 46^{\circ }$, if it is being given that $1^{\circ = 0.01745$ radians.

• A. $1.09$
• B. $2.01$
• C. $1.03$
• D. $1.001$

Answer 1

Answer: (C)

Let $y=f\left(x\right)=tanx,x=45^{\circ }={\left(\frac{\pi }{4}\right)}^c$
and $x+\Delta x=46^{\circ }$.
Then,
$\Delta x=1^{\circ }=0.01745$ radians
For $x=\frac{\pi }{4}$, we have $y=f\left(\frac{\pi }{4}\right)=tan\frac{\pi }{4}=1$
Let $dx=\Delta x=0.01745$
Now, $y=tanx$
$\Rightarrow \frac{dy}{dx}=se{c}^{2}x$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{x=\pi /4}=se{c}^2\frac{\pi }{4}=2$
$\therefore dy=\frac{dy}{dx}dx$
$\Rightarrow dy=2\left(0.01745\right)=0.03490$
$\Rightarrow \Delta y=0.03490$
$\left(\because \Delta y\cong dy\right)$
Hence, $tan46^{\circ }=y+\Delta y$
$=1+0.03490$
$=1.03490\approx \mathrm{1.03.}$