Problem Set 2.4 Derivatif Trigonometri

catatan :

$ \frac{d}{dx} sin x~= cos\, x$

$ D_x (cos x)~= - sin\, x$

$ \frac{d}{dx}tanx~=sec^2 x$

$D_x (cot\, x)~=-csc\, x $

$\frac{d}{dx} sec x~=sec\, x\, tan\, x$ 

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$ 6. ~y = csc\, x$

$ D_x(csc \, x) = D_x \frac{1}{sin x}$

$ = \frac{sin x\, D_x(1) - (1)\, D_x(sin x)}{sin^2 x}$

$ = \frac{0 - cos\, x}{sin^2 x}~=\frac{-1}{sin x}.\frac{cos\, x}{sin\, x}~= -csc\,x \,cot\, x$

$10.~y= \frac{sin x + cos x}{tan x} $ 

$\frac{d}{dx}(\frac{sin x + cos x}{tan x})$

$=\frac{tanx\, D_x (sin x+ cos x) - (sin x+ cos x)\, D_x (tan x)}{tan^2 x}$

 $= \frac{tanx(cosx-sinx)-(sinx+cosx)sec^2 x}{tan^2x}$

$= \frac{tanx(cosx-sinx)-(sinx+cosx)sec^2 x}{\frac{sin^2x}{cos^2x}}$ 

$ = [sin x - \frac{sin^2 x}{cos x} - \frac{sin x}{cos^2 x} - \frac{1}{cos x}]  : [\frac{sin^2 x}{cos^2 x}]$

$ = [sin x - \frac{sin^2 x}{cos x} - \frac{sin x}{cos^2 x} - \frac{1}{cos x}]  [\frac{cos^2 x}{sin^2 x}]$

$ =\frac{cos^2 x}{sin x} - cos x - \frac{1}{sin x} - \frac{cos x}{sin^2 x} $

Problem Set 1.4 : Limit Involving Trigonometric Functions

$ 7.  \lim_{\theta \to 0} \frac{sin\, 3\theta}{tan \, \theta} $

$= \lim_{\theta \to 0} \frac{sin \, 3\theta}{\frac{sin \theta}{cos \theta}} $

$ = \lim_{\theta \to 0} \frac{cos\theta \, sin 3\theta}{sin \theta} $

$ = \lim_{\theta \to 0} [ \frac{sin 3\theta}{3 \theta}. 3\, cos \theta . \frac{1}{sin \theta} . \theta ] $ 

$ = 3 \lim_{\theta \to 0} [ \frac{sin 3\theta}{3\theta}. cos \theta. \frac{\theta}{sin \theta}] $

$ = 3 (1.1.1)~=~3 $

$ 8.  \lim_{\theta \to 0} \frac{tan 5\theta}{sin 2\theta} $

$ = \lim_{\theta \to 0} \frac{\frac{sin 5\theta}{cos 5\theta}}{sin 2\theta} $

$ = lim_{\theta \to 0} \frac{sin5 \theta}{ cos 5\theta \, sin 2\theta} $

$ = lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}.5\theta. \frac{1}{2\theta}] $

$ = lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}.\frac{5}{2}]  $

$ =  \frac{5}{2} lim_{\theta \to 0} [\frac{sin 5\theta}{5\theta}.\frac{2\theta}{sin 2\theta}.\frac{1}{cos 5\theta}] $

$ = \frac{5}{2} (1.1.1)~=~ \frac{5}{2} $

 

$9. \lim_{\theta \to 0} \frac{cot\, \pi \theta \, sin \theta}{2 sec \theta}$

$ = \lim_{\theta \to 0} \frac{\frac{cos \pi \theta}{sin \pi \theta} sin \theta}{\frac{2}{cos \theta}}$

$ = \lim_{\theta \to 0} \frac{cos \pi \theta \, sin \theta \, cos \theta}{2 sin \pi \theta} $

$= \lim_{\theta \to 0} [\frac{sin \theta}{\theta}. \frac{\pi \theta}{sin \pi \theta}. \frac{1}{\pi}. \frac{cos \pi \theta \, cos \theta}{2}]$

$ = \frac{1}{2\pi} ~\lim_{\theta \to 0}  [\frac{sin \theta}{\theta}. \frac{\pi \theta}{sin \pi \theta}. cos \pi \theta \, cos \theta ]$

$ =\frac{1}{2 \pi}(1.1.1) ~=~\frac{1}{2 \pi}$

Dua Gambar Bersisian Pada LaTeX

 %--------DUA GAMBAR BERSISIAN

\begin{figure}

  \begin{minipage}{0.4\textwidth}

    \centering

    \includegraphics[width=\linewidth]{topologi-1-2023.drawio.png}

    \caption{Topologi Mesh 1 singgle Hop}

    \label{fig:topologi1}

  \end{minipage}%

  \begin{minipage}{0.5\textwidth}

    \centering

    \includegraphics[width=\linewidth]{topologi-2-2023.drawio.png}

    \caption{topologi 2 multiHop}

    \label{fig:topologi2}

  \end{minipage}

\end{figure}