Value of tan2°(Without using calculator)
Yesterday my sir asked us a question:"How can you find the value of tan2° without using the calculator? " I asked, whether he is asking the formula of tan 2A or something, but he said no its tan 2°. I tried my head out in every possible way even tried out the approximation method of differentiation, but didn't got any idea. May be it will be something like tan (60°/30°) or something like that, but I get no clue. The exact value is 0.035, but that's coming from a calculator. How to find ourselves the value? Any idea? And I'm not familiar with MathJack so would be grateful if someone edit it out for me. Thanks in advance!
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$\begingroup$The linear approximation formula says that $f(x+\Delta x)\approx f(x)+f'(x)\Delta x$
Let $y=f(x)=\tan x$. Set $x=0$ and $\Delta x=2^{\circ}=\frac{\pi}{90}$ radians.
$\therefore f(0+2^\circ)\approx f(0)+f'(0)\frac{\pi}{90}$
$\implies f(2^\circ)\approx \tan 0+\sec^20\times\frac{\pi}{90}$
i.e., $\tan 2^\circ\approx\frac{\pi}{90}\approx 0.0349.$
Note that we can exclude the approximation of $\frac{\pi}{90}$ which would require a calculator.
$\endgroup$ 3 $\begingroup$Okay so it seems I have found it...
I hope all know for very small angles $\sin\theta$ and $\tan\theta$ become nearly equal to $\theta$, in radians. I hope I don't need to prove that.
So since $2^\circ=2\cdot 0.0175$ radians, or $2^\circ=\frac {2\pi}{180}$,
we get that $\tan(2^{\circ})$ is approximately equal to $0.0349.$
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