Pembuktian Turunan Trigonometri

Masih membahas pembuktian turunan, kali ini kita akan buktikan turunan dari fungsi-fungsi trigonometri

  1. \frac{d}{dx}\sin x=\cos x
  2. \frac{d}{dx}\cos x=-\sin x
  3. \frac{d}{dx}\tan x=\sec^{2}x

Yang pertama

{\displaystyle \frac{d}{dx}\sin x=\lim_{h\rightarrow0}\frac{\sin\left(x+h\right)-\sin x}{h}}

Gunakan penjumlahan sudut pada sin

{\displaystyle =\lim_{h\rightarrow0}\frac{\sin x\cos h+\cos x\sin h-\sin x}{h}}

Urutkan ulang

{\displaystyle =\lim_{h\rightarrow0}\frac{\sin x\cos h-\sin x+\cos x\sin h}{h}}

{\displaystyle =\lim_{h\rightarrow0}\frac{\sin x\left(\cos h-1\right)+\cos x\sin h}{h}}

Gunakan sifat penjumlahan limit

{\displaystyle =\lim_{h\rightarrow0}\frac{\sin x\left(\cos h-1\right)}{h}+\lim_{h\rightarrow0}\frac{\cos x\sin h}{h}}

{\displaystyle =\sin x\lim_{h\rightarrow0}\frac{\left(\cos h-1\right)}{h}+\cos x\lim_{h\rightarrow0}\frac{\sin h}{h}}

Diketahui

{\displaystyle \lim_{h\rightarrow0}\frac{\left(\cos h-1\right)}{h}=0} (Bukti) dan {\displaystyle \lim_{h\rightarrow0}\frac{\sin h}{h}=1} (bukti)

Diperoleh

=\sin x\times0+\cos x\times1=\cos x

Selanjutnya

{\displaystyle \frac{d}{dx}\cos x=\frac{d}{dx}\sin\left(\frac{\pi}{2}-x\right)}

Gunakan aturan rantai

{\displaystyle =-\cos\left(\frac{\pi}{2}-x\right)=-\sin x}

Yang ketiga

{\displaystyle \frac{d}{dx}\tan x=\frac{d}{dx}\frac{\sin x}{\cos x}}

Gunakan aturan pembagian

{\displaystyle =\frac{\cos x\frac{d}{dx}\sin x-\sin x\frac{d}{dx}\cos x}{\cos^{2}x}}

{\displaystyle =\frac{\cos^{2}x+\sin^{2}x}{\cos^{2}x}=\frac{1}{\cos^{2}x}=\sec^{2}x}

Fungsi Trigonometri Lainnya

Untuk pembuktian turunan dari sec, cosec dan cot, kita gunakan aturan rantaia

{\displaystyle \frac{d}{dx}\csc x=\frac{d}{dx}\sin^{-1}x=-\sin^{-2}x\cos x=-\sin^{-1}x\cot x=-\csc x\cot x}

{\displaystyle \frac{d}{dx}\sec x=\frac{d}{dx}\cos^{-1}x=\cos^{-2}x\sin x=\cos^{-1}x\tan x=\sec x\tan x}

{\displaystyle \frac{d}{dx}\cot x=\frac{d}{dx}\tan^{-1}x=-\tan^{-2}x\sec^{2}x=-\csc^{2}x}

 

 

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About Aria Turns

Seorang Alumnus Matematika UGM, dengan ilmu yang didapat ketika kuliah (Padahal sering bolos kuliah :p ), saya menyebarkan virus matematika
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2 Responses to Pembuktian Turunan Trigonometri

  1. Tugas q terselesaikan… Mksh .Oya di tunggu kunjungannya ya

  2. aryasuta andhikacipta says:

    Edan pak susah amet

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