Penjumlahan hasil sinus dan cosinus

Posted on December 30, 2015 by Nursatria

Kembali kita bermain Trigonometri, kita akan membuktikan penjumlahan hasil sinus dan cosinus

  1. {\displaystyle \sin\theta+\sin\phi=2\sin\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)}
  2. {\displaystyle \sin\theta-\sin\phi=2\sin\left(\frac{\theta-\phi}{2}\right)\cos\left(\frac{\theta+\phi}{2}\right)}
  3. {\displaystyle \cos\theta+\cos\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)}
  4. {\displaystyle \cos\theta-\cos\phi=-2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right)}

Bukti:

Pertama-tama kita mengunakan sifat penjumlahan sudut

\sin\left(A+B\right)=\sin A\cos B+\cos A\sin B

\sin\left(A-B\right)=\sin A\cos B-\cos A\sin B

Jumlahkan keduanya

\sin\left(A+B\right)+sin\left(A-B\right)=\sin A\cos B+\cos A\sin B+\sin A\cos B-\cos A\sin B=2\sin A\cos B

Selanjutnya kurangi keduanya:

\sin\left(A+B\right)-sin\left(A-B\right)=\sin A\cos B+\cos A\sin B-\sin A\cos B+\cos A\sin B=2\cos A\sin B

Subtitusi \theta=A+B dan \phi=A-B yang berarti {\displaystyle A=\frac{\theta+\phi}{2}} dan {\displaystyle B=\frac{\theta-\phi}{2}}, diperoleh

{\displaystyle \sin\theta+\sin\phi=2\sin\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)}

{\displaystyle \sin\theta-\sin\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right)=2\sin\left(\frac{\theta-\phi}{2}\right)\cos\left(\frac{\theta+\phi}{2}\right)}

Serupa untuk cosinus, pertama-tama kita gunakan penjumlahan sudut

\cos\left(A+B\right)=\cos A\cos B-\sin A\sin B

\cos\left(A-B\right)=\cos A\cos B+\sin A\sin B

Kembali lakukan penjumlahan dan pengurangan pada keduanya

\cos\left(A+B\right)+\cos\left(A-B\right)=2\cos A\cos B

\cos\left(A+B\right)-\cos\left(A-B\right)=-2\sin A\sin B

Subtitusi \theta dan \phi seperti diatas diperoleh

{\displaystyle \cos\theta+\cos\phi=2\cos\left(\frac{\theta+\phi}{2}\right)\cos\left(\frac{\theta-\phi}{2}\right)}

{\displaystyle \cos\theta-\cos\phi=2\sin\left(\frac{\theta+\phi}{2}\right)\sin\left(\frac{\theta-\phi}{2}\right)}

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About Nursatria

Seorang Alumnus Matematika UGM, dengan ilmu yang didapat ketika kuliah (Padahal sering bolos kuliah :p ), saya menyebarkan virus matematika
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1 Response to Penjumlahan hasil sinus dan cosinus

  1. Dyvictory says:
    April 6, 2016 at 15:02

    Blog ini asik banget buat belajar… apalagi kelemahan saya selain di logaritma juga sincostan… kalo nggak ngerti, larinya ke sini ah XD

    Reply

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