diposting : Apriwida Yanti (6)
Blog IPA 6 SMANIK
Soal Trigonometri :

Buktikan bahwa :
y=sin\;x\;\;\;maka\;\;\;y\;'=cos\;x  dan



y=cos\;x\;\;\;maka\;\;\;y\;'=-sin\;x 

Jawaban :

Kita pakai rumus turunan
f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}
diketahui

f(x)=sin\;x
f(x+h)=sin\;(x+h)
ingat rumus
{\color{Red} sin\;A-sin\;B=2.cos\;\frac 12(A+B).sin\frac 12(A-B) }
maka
\begin{align*} sin(x+h)-sin\;x&=&2.cos\;\frac 12(x+h+x).sin\frac 12(x+h-x)\\&=&2.cos\frac 12(2x+h).sin\frac 12h\end{align*}
Kita subtitusikan ke rumus turunan
\begin{align*}f'(x)&=&\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}\\&=&\lim_{h \to 0}\frac{sin\;(x+h)-sin\;x}{h}\\&=&\lim_{h \to 0}\frac{2.cos\frac 12(2x+h).sin\frac 12h}{h}\\&=&\lim_{h \to 0}2.cos\frac 12(2x+h).\lim_{h \to 0}\frac{sin\frac 12h}{h}\\&=&\lim_{h \to 0}2.cos\frac 12(2x+h).\frac 12\\&=&2.cos\frac 12(2x+0).\frac 12\\&=&2.cos\;x.\frac 12\\f'(x)&=&{\color{Red} cos\;x}\end{align*}
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