Question

If Sin alpha =3/5 ,find Tan 3 alpha

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{tan\:3\alpha=\frac{117}{28}}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies sin \: \alpha = \frac{3}{5} \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies tan \: 3 \alpha =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies sin \: \alpha = \frac{3}{5} - - - - - (1) \\ \\ \tt: \implies sin \: \alpha = \frac{p}{h} - - - - - (2) \\ \\ \text{from \: (1) \: and \: (2)} \\ \\ \green{ \tt: \implies p = 3} \\ \\ \green{\tt: \implies h = 5 }\\ \\ \bold{As \: we \: know \: that} \\ \tt: \implies {h}^{2} = {p}^{2} + {b}^{2} \\ \\ \tt: \implies {5}^{2} = {3}^{2} + {b}^{2} \\ \\ \tt: \implies 25 - 9 = {b}^{2} \\ \\ \tt: \implies b = \sqrt{16} \\ \\ \green{\tt: \implies b = 4} \\ \\ \bold{For \: finding \: value } \\ \tt: \implies tan \: 3 \alpha = \frac{3 \: tan \: \alpha - {tan}^{3} \: \alpha }{1 - 3 \: {tan}^{2} \: \alpha }$

$\tt: \implies tan \: 3 \alpha = \frac{3 \times \frac{p}{b} - {( \frac{p}{b} })^{3} }{1 - 3 \times (\frac{p}{b})^{2} } \\ \\ \tt: \implies tan \: 3 \alpha = \frac{3 \times \frac{3}{4} - (\frac{3}{4})^{3} }{1 - (\frac{3}{4})^{2} } \\ \\ \tt: \implies tan \: 3 \alpha = \frac{ \frac{9}{4} - \frac{27}{64} }{1 - \frac{9}{16} } \\ \\ \tt: \implies tan \: 3 \alpha = \frac{ \frac{144 - 27}{64} }{ \frac{16 - 9}{16} } \\ \\ \tt: \implies tan \: 3 \alpha = \frac{117}{64} \times \frac{16}{7} \\ \\ \green{\tt: \implies tan \: 3 \alpha = \frac{117}{28} }$