Answer :
[tex]tan\ 20+4\ sin\ 20\\ \\ = \frac{sin\ 20}{ cos 20} + 4\ sin\ 20 \\ \\ = \frac{sin\ 20+4\ sin\ 20\ cos\ 20}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ sin\ 40}{cos\ 20} \\ \\ = = \frac{sin\ 20+2\ sin\ (60-20)}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ sin\ 60\ cos\ 20 - 2\ sin\ 20\ cos\ 60}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ ( \sqrt{3}/2 )\ cos\ 20 - 2\ sin\ 20\ (1/2)}{cos\ 20} \\ \\ = \frac{sin\ 20+ \sqrt{3}\ cos\ 20 -sin\ 20}{cos\ 20} \\ \\ = \frac{ \sqrt{3}\ cos\ 20 }{cos\ 20}\\ \\ = \boxed{ \sqrt{3} } [/tex]
Step-by-step explanation:
\begin{lgathered}tan\ 20+4\ sin\ 20\\ \\ = \frac{sin\ 20}{ cos 20} + 4\ sin\ 20 \\ \\ = \frac{sin\ 20+4\ sin\ 20\ cos\ 20}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ sin\ 40}{cos\ 20} \\ \\ = = \frac{sin\ 20+2\ sin\ (60-20)}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ sin\ 60\ cos\ 20 - 2\ sin\ 20\ cos\ 60}{cos\ 20} \\ \\ = \frac{sin\ 20+2\ ( \sqrt{3}/2 )\ cos\ 20 - 2\ sin\ 20\ (1/2)}{cos\ 20} \\ \\ = \frac{sin\ 20+ \sqrt{3}\ cos\ 20 -sin\ 20}{cos\ 20} \\ \\ = \frac{ \sqrt{3}\ cos\ 20 }{cos\ 20}\\ \\ = \boxed{ \sqrt{3} }\end{lgathered}tan 20+4 sin 20=cos20sin 20+4 sin 20=cos 20sin 20+4 sin 20 cos 20=cos 20sin 20+2 sin 40==cos 20sin 20+2 sin (60−20)=cos 20sin 20+2 sin 60 cos 20−2 sin 20 cos 60=cos 20sin 20+2 (3/2) cos 20−2 sin 20 (1/2)=cos 20sin 20+3 cos 20−sin 20=cos 203 cos 20=√3
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