3. Rumus Tangen Jumlah dan Selisih Dua Sudut
\(\tan (A + B)\) = \(\frac{\sin (A + B)}{\cos (A + B)}\)
= \(\frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B}\)
= \(\frac{\sin A \cos B + \cos A \sin B}{\cos A \cos B - \sin A \sin B} \cdot \frac{\frac{1}{\cos A \cdot \cos B}}{\frac{1}{\cos A \cdot \cos B}}\)
= \(\frac{\frac{\sin A \cos B + \cos A sin B}{\cos A \cos B}}{\frac{\cos A \cos B - \sin A \sin B}{\cos A \cos B}}\)
= \(\frac{\frac{\sin A \cos B}{\cos A \cos B} + \frac{\cos A \sin B}{\cos A \cos B}}{\frac{\cos A \cos B}{\cos A \cos B} - \frac{\sin A \sin B}{\cos A \cos B}}\)
= \(\frac{\frac{\sin A}{\cos A}+\frac{\sin B}{\cos B}}{1- \frac{\sin A}{\cos B} \cdot \frac{\sin B}{\cos B}}\) = \(\frac{\tan A + \tan B}{1 - \tan A \tan B}\)
Rumus tangen jumlah dua sudut:
\(\tan (A + B)\) = \(\frac{\tan A + \tan B}{1 - \tan A \tan B}\)
\(\tan (A - B)\) = \(\frac{\tan A - \tan B}{1 + \tan A \tan B}\)
Contoh soal
Tanpa menggunakan tabel logaritma atau kalkulator, hitunglh \(\tan 105^\circ\)
Penyelesaian
\(\tan 105^\circ\) = \(\tan (60 + 45)^\circ)\) = \(\frac{tan 60^\circ + \tan 45^\circ}{1 - \tan 60^\circ \tan 45^\circ}\)
= \(\frac{\sqrt{3}+1}{1- \sqrt{3}}\) = \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}} x \frac{1 + \sqrt{3} }{1 + \sqrt{3}} \)
= \(\frac{\sqrt{3}+3+1+\sqrt{3}}{1^2-(\sqrt{3})^2}\) = \(\frac{4 + 2\sqrt{3}}{1- 3}\) = \(\frac{4 + 2\sqrt{3}}{-2}\) = \(-(2 + \sqrt{3}\)
Rumus Tangen Jumlah dan Selisih Dua Sudut
Selesai