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SMK Teknologi dan Rekayasa

Indikator UN 2011 Matematika SMK Teknologi (24)


Indikator 24
Menentukan nilai perbandingan trigonometri selisih dua sudut bila diketahui perbandingan trigonometri sinus dan tangen

Ditentukan\ sin\ A=\frac{5}{13}\ dan\ tan\ B=\frac{3}{4},\ maka\ nilai\ cos\ (A-B)= .... \\\\a.\ \ \frac{20}{65}\\\\b.\ \ \frac{24}{36}\\\\c.\ \ \frac{33}{65}\\\\d.\ \ \frac{56}{65}\\\\e.\ \ \frac{63}{65}
Jawaban: e
cos\ (A-B)=cos\ A\ cos\ B+sin\ A\ sin\ B
sin\ A=\frac{5}{13}=\frac{\bf de}{\bf mi}\quad \rightarrow \quad cos\ A=\frac{12}{13} (Ingat, Triple Bilangan Pythagoras: 5, 12, 13)
tan\ B=\frac{3}{4}=\frac{\bf de}{\bf sa}
Dengan mengingat Triple Bilangan Pythagoras: 3, 4, 5 diperoleh:
cos\ B=\frac{\bf sa}{\bf mi}=\frac{4}{5}\quad dan\quad sin\ B=\frac{\bf de}{\bf mi}=\frac{3}{5}
Jadi:
\begin{array}{rcl}cos\ (A-B)&=&\big(\frac{12}{13}\times \frac{4}{5}\big)+\big(\frac{5}{13}\times \frac{3}{5}\big)\\\\&=&\frac{48}{65}+\frac{15}{65}\\\\&=&\frac{63}{65}\end{array}

Keterangan:
de = panjang sisi di depan sudut
sa = panjang sisi di samping sudut
mi = panjang sisi miring (hypotenuse)

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