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I am Not Asking For Like 30° or 60°... I am Asking For Angles Like 36° , 22.5° ... etc Something Like These...

For Example :

tan 22.5° = √2 - 1

sin 18° = (√5 -1)/4

I am Not Asking For The Values, Just The Important Angles Values I Can Take Out By Myself

Just Which Angles Do I Need To Remember
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@Saikat Das, @peter petrov's opinion is indisputable. Also, myself being an IIT aspirant, I can understand your concern. As you mentioned in the comment that there are $1.4\,\text{m}$ candidates appearing the exam but the toughest competitor you've to beat is time. In order to solve the questions in given time you have to fast and agile which in turn requires memorization of few formulae and results.

The renowned author of mathematics for JEE, Dr.R.D. Sharma, in his book "Objective Mathematics Vol.II" has given trigonometric ratios of some important angles which are useful for IIT-JEE. Here are they: (I've mentioned only $\sin$ of the angles)

  1. $\sin 15^\circ=\displaystyle\frac{\sqrt 3 -1}{2\sqrt 2}$

  2. $\sin 22.5^\circ=\displaystyle\frac{1}{2}(\sqrt{2-\sqrt 2})$

  3. $\sin 18^\circ =\displaystyle\frac{\sqrt 5-1}{4}$

  4. $\sin 36^\circ=\displaystyle\frac{\sqrt{10+2\sqrt 5}}{4}$

  5. $\sin 9^\circ=\displaystyle\frac{\sqrt{3+\sqrt 5}-\sqrt{5-\sqrt 5}}{4}$

  6. $\sin 37^\circ\approx \displaystyle\frac{3}{5}$

  7. $\cos 36^\circ-\cos 72^\circ=\displaystyle\frac{1}{2}$

  8. $\cos 36^\circ\cos 72^\circ=\displaystyle\frac{1}{4}$

NOTE: This results are written just for completeness sake. If you check out past year papers they mostly ask trig ratios of standard angles and $37^\circ$. In your question you said that you can find the values by yourself. I suggest don't try except for $15^\circ$ and $22.5^\circ$ (as they are half of standard angles) because it is time consuming and not worth it.

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  • $\begingroup$ @Integrand, I know that, it is approximated. JEE uses $\sin 37^\circ=3/5$ and $\cos 37^\circ=4/5$ for simplicity. $\endgroup$ – SarGe Jul 6 at 13:18
  • $\begingroup$ Also remember cot (9.5)= root 2+ root 3 root 4 + root 6 $\endgroup$ – Anshaj Shukla Jul 6 at 18:32
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I've never tried to remember the values of the trigonometric
functions for any angles other than $30°, 60°, 45°, 135°$.

I would be surprised if any math exam really requires memorizing
any such values just because math is not really about that.

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    $\begingroup$ Its Just For My Benefit To , Solve Quesion At a Faster Rate... You See in IIT-JEE Exam There Are About 1.4 Million Candidates, And Only 10K are Selected... Because Of This Much High Competition, I Was Asking Some Obvious Degrees Asked In Phy, Or Maths Problems... $\endgroup$ – Saikat Das Jul 6 at 11:05

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