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

The value of sin(2arctan(3)) is 3/5

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sin(2arctan(3))

sin (2 arctan (3))

\frac{3}{5}

Solution steps

sin (2 arctan (3))

Rewrite using trig identities:

2 sin (arctan (3)) cos (arctan (3))

= 2 sin (arctan (3)) cos (arctan (3))

Rewrite using trig identities:

sin (arctan (3)) = \frac{3 10}{10}

Rewrite using trig identities:

cos (arctan (3)) = \frac{10}{10}

= 2 \cdot \frac{3 10}{10} \cdot \frac{10}{10}

Simplify

2 \cdot \frac{3 10}{10} \cdot \frac{10}{10} : \frac{3}{5}

= \frac{3}{5}

\frac{1}{cos ^{2} ( 4 0 ^{\circ} )} - \frac{1}{cot ^{2} ( 4 0 ^{\circ} )}

\frac{9}{sin ( 3 0 ^{\circ} )}

370 \cdot \frac{1}{cos ( 2 0 ^{\circ} )} \cdot sin (2 0^{\circ})

2 sin (\frac{2 π}{9}) cos (\frac{2 π}{9})

- cos (2 π) + cos (π)