Jika cos (A−B)=53 dan tan A tan B=2, hitunglah: c. cos(A+B)

Pertanyaan

Jika $\cos\;\left(A-B\right)=\frac35$ dan $\tan\;A\;\tan\;B=2$ , hitunglah:

c. $\cos\left(A+B\right)$

D. Rajib

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Malang

Jawaban terverifikasi

Jawaban

diperoleh $\cos\left(A+B\right)=-\frac15$ .

Pembahasan

Diketahui $\cos\left(A-B\right)=\frac35$ dan $\tan\left(A\right)\tan\left(B\right)=2$ . Akan dicari $\cos\left(A+B\right)$

Ingat bahwa

$\cos(A-B)=\cos(A)\cos(B)+\sin(A)\sin(B)\\\cos(A+B)=\cos(A)\cos(B)-\sin(A)\sin(B)$

Diperhatikan

$tan open parentheses A close parentheses times tan open parentheses B close parentheses equals 2 rightwards double arrow fraction numerator sin left parenthesis A right parenthesis over denominator cos left parenthesis A right parenthesis end fraction times fraction numerator sin left parenthesis B right parenthesis over denominator cos left parenthesis B right parenthesis end fraction equals 2 rightwards double arrow sin left parenthesis A right parenthesis times sin left parenthesis B right parenthesis equals 2 space cos left parenthesis A right parenthesis times cos left parenthesis B right parenthesis$

dan

$tan open parentheses A close parentheses times tan open parentheses B close parentheses equals 2 rightwards double arrow fraction numerator sin left parenthesis A right parenthesis over denominator cos left parenthesis A right parenthesis end fraction times fraction numerator sin left parenthesis B right parenthesis over denominator cos left parenthesis B right parenthesis end fraction equals 2 rightwards double arrow fraction numerator sin left parenthesis A right parenthesis times sin left parenthesis B right parenthesis over denominator 2 end fraction equals cos left parenthesis A right parenthesis times cos left parenthesis B right parenthesis$

Diperoleh perhitungan

$\begin{array}{rcl}{\cos(A-B)}&=&\frac35\\{\cos(A)\cos(B)+\sin(A)\sin(B)}&=&\frac35\\{\cos(A)\cos(B){+2\cos(A)\cos(B)}}&=&\frac35\\{3\cos(A)\cos(B)}&=&\frac35\\{\cos(A)\cos(B)}&=&\frac35\cdot\frac13\\{\cos(A)\cos(B)}&=&\frac15\end{array}$

dan

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos left parenthesis A minus B right parenthesis end cell equals cell 3 over 5 end cell row cell cos left parenthesis A right parenthesis cos left parenthesis B right parenthesis plus sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis end cell equals cell 3 over 5 end cell row cell fraction numerator sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis over denominator 2 end fraction plus sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis end cell equals cell 3 over 5 end cell row cell 3 over 2 sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis end cell equals cell 3 over 5 end cell row cell sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis end cell equals cell 3 over 5 times 2 over 3 end cell row cell sin left parenthesis A right parenthesis sin left parenthesis B right parenthesis end cell equals cell 2 over 5 end cell end table$

Lebih lanjut, diperoleh

$\begin{array}{rcl}\cos(A+B)&=&\cos(A)\cos(B)-\sin(A)\sin(B)\\&=&\frac15-\frac25\\&=&-\frac15\end{array}$

Dengan demikian, diperoleh $\cos\left(A+B\right)=-\frac15$ .