<s>[INST] Calculate the approximate value I am supposed to calculate the approximate value of $$\cos 151^\circ$$ My idea was that I can divide it in the form: $$\cos 90^\circ+ 61^\circ= \cos \frac{\pi}{2}+ \left (\frac{\pi}{3} +\frac{\pi}{180} \right )$$ Then I use the derivation for cosx:
$$-\sin \frac{\pi}{2}\left ( \frac{\pi}{3} +\frac{\pi}{180} \right )=-\frac{61\pi}{180}+61^\circ$$
But I guess, it is not correct.
Can anyone help me?
 [/INST] One way is to use derivatives.
Let $y=f(x) = \cos x$
$dy = -\sin x dx$
Let $x=150^\circ = \frac{2\pi}{3}  , dx = 1^\circ\approx 0.0174$ 
$\cos x = \cos150^\circ \approx -0.8660, \sin x = \sin 150^\circ =0.5$ 
$y+dy = f(x+dx) = \cos x -\sin x dx = \cos(150^\circ)-\sin(150^\circ)\times0.0174 \approx-0.8747$
$\implies\cos(151^\circ) \approx -0.8747$
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