Linear approximation of cos(28 degrees)

Question

Evaluate cos(28 degrees) using linear approximation.

I have done this so far, but the answer seems to be incorrect and I can't figure out why.

y= 30

f(x) = cos(x)
f'(x) = -sin(x)

f(y) = f(30) = cos(30) = sqrt(3)/2
f(y) = f(30) = -sin(30) = -1/2

f(x) = f(y) + f(y) (x-y)
 = sqrt(3)/2 + -1/2 (28-30)
 = sqrt(3)/2 + 1
 = 1.866025404

@AméricoTavares Sorry, i don't get it. Is it possible to explain to me? Thank you :) — Phantom, Feb 20 '14 at 12:36
Not $30$ but $30 \pi/180$ would be better. Same for $28$ to be $28 \pi/180$ and you are done. — Claude Leibovici, Feb 20 '14 at 12:36
Please see AnonSubmitter85's answer. If $x$ is in radians, then the angle in degrees is $\frac{\pi }{180}x$ and for $f(x)=\cos \frac{\pi }{180}x$, we have that $f^{\prime }(x)=-\frac{\pi }{180}\sin \frac{\pi }{180}x$. — Américo Tavares, Feb 20 '14 at 12:42

score 3 · Accepted Answer · answered Feb 20 '14 at 12:36

3

You are forgetting to account for unit conversion. What you have is $f(x) = \cos( \pi/180 \cdot x )$, so $f'(x) = -\sin(\pi/180 \cdot x ) \cdot \pi/180$. Then you will get

$$ f(28) \approx f(30) + f'(30) \cdot (28-30) = \cos( \pi/6 ) - \sin(\pi/6) \cdot \pi/180 \cdot -2 = 0.883478\dots $$

answered Feb 20 '14 at 12:36

AnonSubmitter85

3,455

Typo's occured to me and lead to stupidity ! Sorry. – Claude Leibovici Feb 20 '14 at 12:39
@ClaudeLeibovici Yes. – AnonSubmitter85 Feb 20 '14 at 12:40
@AnonSubmitter85 Thank you! One more question, so if we do a f"(x), will it be -cos(π/180⋅x) ⋅ π/180 ⋅ π/180 or will it be just be cos(π/180⋅x)⋅π/180? – Phantom Feb 20 '14 at 13:03
1

@Phantom. Please, forget degrees. You would have a lot of problems if you don't. Cheers. – Claude Leibovici Feb 20 '14 at 13:11
1

@Phantom You would still need to apply the chain rule, so it would be $-\pi^2/180^2 \cdot \cos( \pi/180 \cdot x)$. But Claude is right when he says you should be using radians instead. – AnonSubmitter85 Feb 20 '14 at 13:15
@ClaudeLeibovici Hahaha...okay. As far as maths is concerned, i always get confused. Thank you very much for the advice :D – Phantom Feb 20 '14 at 13:19
@AnonSubmitter85 Thank you very much! It is much more clearer now :) – Phantom Feb 20 '14 at 13:19

Linear approximation of cos(28 degrees)

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