Question

Why does sin(x) only differentiate to cos(x) if x is defined in radians and not in degrees?

Answer 1

We know the standard equation for the derivative of a function f(x).

$\displaystyle\dfrac {d}{dx}f(x)=\lim_{h\to0}\dfrac {f(x+h)-f(x)}{h}$

Here we are going to find the derivative of sin(x). So we have,

$\displaystyle\dfrac {d}{dx}\sin(x)=\lim_{h\to0}\dfrac {\sin(x+h)-\sin(x)}{h}\quad\longrightarrow\quad (1)$

But, by sum - to - product formula,

$\sin(x+h)-\sin (x)=2\cos\left(\dfrac {x+h+x}{2}\right)\sin\left (\dfrac {x+h-x}{2}\right)\\\\\\\sin(x+h)-\sin (x)=2\cos\left(x+\dfrac {h}{2}\right)\sin\left (\dfrac {h}{2}\right)$

Then (1) becomes,

$\displaystyle\dfrac {d}{dx}\sin(x)=\lim_{h\to0}\dfrac {2\cos\left(x+\dfrac {h}{2}\right)\sin\left (\dfrac {h}{2}\right)}{h}\\\\\\\dfrac {d}{dx}\sin(x)=\lim_{h\to0}\cos\left(x+\dfrac {h}{2}\right)\cdot\lim_{h\to0}\dfrac {2\sin\left (\dfrac {h}{2}\right)}{h}\\\\\\\dfrac {d}{dx}\sin(x)=\lim_{h\to0}\cos\left(x+\dfrac {h}{2}\right)\cdot\lim_{h\to0}\dfrac {\sin\left (\dfrac {h}{2}\right)}{\left (\dfrac {h}{2}\right)}\quad\longrightarrow\quad (2)$

But, by the following trigonometric limit identity,

$\displaystyle\lim_{x\to0}\dfrac {\sin (x)}{x}=1$

we can say that,

$\displaystyle\lim_{h\to0}\dfrac {\sin\left (\dfrac {h}{2}\right)}{\left (\dfrac {h}{2}\right)}=1$

Then (2) becomes,

$\dfrac {d}{dx}\sin(x)=\lim_{h\to0}\cos\left(x+\dfrac {h}{2}\right)\cdot1\\\\\\\dfrac {d}{dx}\sin(x)=\cos\left(x+\dfrac {0}{2}\right)\\\\\\\boxed {\dfrac {d}{dx}\sin(x)=\cos(x)}$

That's why sin(x) only differentiate to cos(x) if x is defined in radians.

What about if x is defined in degrees?

We know about the conversion of angle between degree and radian.

$1^{\circ}=\left (\dfrac {\pi}{180}\right)^c$

Thus,

$x^{\circ}=\left (\dfrac {\pi}{180}x\right)^c$

Then,

$\dfrac {d}{dx}\sin(x^{\circ})=\dfrac {d}{dx}\sin\left (\dfrac {\pi}{180}x\right)\quad\longrightarrow\quad (3)$

But the identity says that,

$\dfrac {d}{dx}\sin(nx)=n\cos(nx)$

Then (3) becomes,

$\dfrac {d}{dx}\sin(x^{\circ})=\dfrac {\pi}{180}\cos\left (\dfrac {\pi}{180}x\right)$

So this is the derivative if x is defined in degrees.

#answerwithquality

#BAL

Answer 2

Answer:

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