What is the value of sin(2arctan(2/3)) ?

The value of sin(2arctan(2/3)) is 12/13

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sin(2arctan(2/3))

sin (2 arctan (\frac{2}{3}))

\frac{12}{13}

Solution steps

sin (2 arctan (\frac{2}{3}))

Rewrite using trig identities:

2 sin (arctan (\frac{2}{3})) cos (arctan (\frac{2}{3}))

= 2 sin (arctan (\frac{2}{3})) cos (arctan (\frac{2}{3}))

Rewrite using trig identities:

sin (arctan (\frac{2}{3})) = \frac{2 13}{13}

Rewrite using trig identities:

cos (arctan (\frac{2}{3})) = \frac{3 13}{13}

= 2 \cdot \frac{2 13}{13} \cdot \frac{3 13}{13}

Simplify

2 \cdot \frac{2 13}{13} \cdot \frac{3 13}{13} : \frac{12}{13}

= \frac{12}{13}

- arctan (tan (\frac{2 π}{7}))

\frac{40}{cos ( 3 0 ^{\circ} )}

arctan (\frac{- 4}{5})

- sec (\frac{11 π}{6}) tan (\frac{11 π}{6})

\frac{7}{cos ( 3 5 ^{\circ} )}

What is the value of sin(2arctan(2/3)) ?
The value of sin(2arctan(2/3)) is 12/13