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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!

Aneek
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    For small $x$, $\tan{x} \approx x$, when $x$ is in radians. So $\tan{2}= \approx \frac{2 \pi}{180}$. – Paul Aug 30 '16 at 16:14
  • $\tan 2^\circ$ is not exactly 0.035. That's an approximation. –  Aug 30 '16 at 16:21
  • @Paul yup I got it.... $0.0175*2=0.035$..that's the approx value....Thanks! – Aneek Aug 30 '16 at 16:25
  • @Paul Why not write it up as an answer? I'll upvote your answer. – amWhy Aug 30 '16 at 16:35
  • @amWhy May I write? I hope Paul hasn't started working on it...waiting for response from Paul – Aneek Aug 30 '16 at 16:36
  • @Aneek Sure, you may write up an answer. Go ahead, and I'm sure Paul will understand! Wait maybe 5 minutes or so. – amWhy Aug 30 '16 at 16:38
  • @amWhy, I have no problem with OP writing it. I was just at lunch, so I didn't see the notes here until I got back! – Paul Aug 30 '16 at 17:50

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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 $0.0349.$

amWhy
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Aneek
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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.

StubbornAtom
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  • Nice work...I was thinking of using approximating formula to calculate it, and you came up with it...well done! And yes, if you know the conversion ratio of degree to radian then I guess you don't have to use the calculator at all! – Aneek Sep 04 '16 at 10:24
  • How is 2 degrees = $2\pi/180$ ? One measure in radian, another is in degrees. –  Aug 02 '17 at 06:58
  • @Abcd That's the well known conversion 'formula' so to speak. Did you check out this article on Wikipedia? – StubbornAtom Aug 03 '17 at 15:37