diketahui p-q=45 derajat dan tan p=4/3,hitunglah nilai tan (2p+q)

Kelas : 11
Mapel : Matematika
Kategori : Trigonometri Lanjut
Kata kunci: tan penjumlahan dan pengurangan sudut
Kode: 11.2.3 (Kelas 11 Matematika Bab 3-Trigonometri Lanjut)

Diketahui p-q=45° dan tan p=4/3,hitunglah nilai tan (2p+q)

Pembahasan:

Rumus tan penjumlahan dan pengurangan sudut:
[tex]\tan ( \alpha + \beta )= \frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta } \\\tan ( \alpha - \beta )= \frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta } [/tex]


[tex]\tan (p - q )= \frac{\tan p -\tan q }{1+\tan p \tan q } \\ tan 45^{\circ}= \frac{ \frac{4}{3}-\tan q }{1+ \frac{4}{3}\tan q } \\ 1= \frac{ \frac{4}{3}-\tan q }{1+ \frac{4}{3}\tan q } \\ 1+ \frac{4}{3}\tan q= \frac{4}{3}-\tan q \\ \frac{4}{3}\tan q+\tan q= \frac{4}{3}-1 \\ \frac{7}{3}\tan q = \frac{1}{3} \\ \tan q= \frac{1}{7} [/tex]

[tex]\tan 2 \alpha = \frac{2\tan \alpha }{1-\tan^2 \alpha} \\ \tan 2p= \frac{2\times \frac{4}{3} }{1-( \frac{4}{3} )^2} \\ \tan 2p = \frac{ \frac{8}{3} }{1- \frac{16}{9} } \\ \tan 2p = \frac{ \frac{8}{3} }{- \frac{7}{9} } \\ \tan 2p = \frac{8}{3}\times (- \frac{9}{7} ) \\ \tan 2p = - \frac{24}{7} [/tex]

[tex]\tan (2p+q)= \frac{\tan 2p + \tan q}{1-\tan2p \tan q} \\ = \frac{- \frac{24}{7}+ \frac{1}{7} }{1-(- \frac{24}{7}( \frac{1}{7} ) )} \\ = \frac{- \frac{23}{7} }{1+ \frac{24}{49} } \\ =- \frac{23}{7}\times \frac{49}{73} \\ =- \frac{161}{73} [/tex]

Semangat belajar!
Semoga membantu :)