I need to remember the values of sine and cosine of 18 degrees,36 degrees,54 degrees,72 degrees. That is multiples if 18 degrees.Is it possible to derive them in about a minute or so ? Do you use any particular method to remember them ?
Yes I do know that we can solve the equation $\sin{5\theta}=\pi/2$ and similar ones but that's way too lengthy (during exams).
Best Answer
This is the method I used during my high school days.
Note that $$\sin 18^\circ = \frac{\sqrt5-1}{4}$$ and $$\sin 36^\circ = \sqrt{\frac{5-\sqrt5}{8}}$$
Now you just have to remember this much.
For $\cos 18^\circ$, you will have the same expression as $\sin 36^\circ$ but only with the minus sign replaced by a plus sign.
Similarly,for $\cos 36^\circ$, you will have the same expression as $\sin 18^\circ$ but only with the minus sign replaced by a plus sign.
And we know that $54^\circ$ and $36^\circ$ are complementary just as $72^\circ$ and $18^\circ$ are. So you can calculate them using the rule: $$\sin (90^\circ - x) = \cos x$$ and $$\cos (90^\circ - x) = \sin x$$