Hoofdstuk 12 Differentiëren *: Afgeleiden van standaardfuncties *
Derivatives of trigonometric functions
Trigonometric rules for differentiation
The derivatives of the fundamental trigonometric functions are mentioned below.
| function | derivative |
| #\sin(x)# | #\cos(x)# |
| #\cos(x)# | #-\sin(x)# |
| #\tan(x)# | #\dfrac{1}{\cos(x)^2}# |
We only deal with a derivation of the first row, in which it is stated that the derivative of #\sin# is equal to #\cos#. The Addition formulas for trigonometric formulas give the following two equalities.
\[\begin{array}{rcl}\sin (x + h) &=& \sin\left(\left(x + \dfrac{h}{2}\right)+ \dfrac{h}{2}\right) \\ &=&\sin\left(x + \dfrac{h}{2}\right)\cos\left(\dfrac{h}{2}\right) + \cos\left(x + \dfrac{h}{2}\right)\sin\left(\dfrac{h}{2}\right) \\ \sin (x) &=&\sin\left(\left(x + \dfrac{h}{2}\right)- \dfrac{h}{2}\right) \\ &=&\sin\left(x + \dfrac{h}{2}\right)\cos\left( \dfrac{h}{2}\right) - \cos\left(x + \dfrac{h}{2}\right)\sin\left( \dfrac{h}{2}\right)\end{array}\]
If we subtract these equalities from each other, we get \[\dfrac{\sin (x + h) −\sin (x)}{h} = \dfrac{2\cos\left(x+\dfrac{h}{2}\right)\sin\left(\dfrac{h}{2}\right)}{h} = \cos\left(x+\dfrac{h}{2}\right)\dfrac{\sin\left(\dfrac{h}{2}\right)}{\dfrac{h}{2}}\tiny.\] If we take the limit for #h\to0# and use the limit #\lim_{t\to0}\dfrac{\sin(t)}{t} =1#, we find: \[\begin{array}{rcl}\sin'(x) &=&\lim_{h\to0}\dfrac{\sin (x + h) −\sin (x)}{h} \\ &=& \lim_{h\to0} \cos\left(x+\dfrac{h}{2}\right) \lim_{h\to0} \dfrac{\sin(\dfrac{h}{2}) }{\dfrac{h}{2}}\\ &=& \cos(x)\cdot 1 \\ &=& \cos(x)\tiny.\end{array}\]
This leads to #\sin' = \cos#.
The derivative of #\tan# is also known as #\sec^2# (see Tangent and cotangent).
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