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}

Decimal Notation

0.6

Solution steps

sin (2 arctan (3))

Rewrite using trig identities:

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

sin (2 arctan (3))

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 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}

sin (arctan (3))

Rewrite using trig identities:

sin (arctan (3)) = \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

Use the following identity:

sin (arctan (x)) = \frac{x 1 + x ^{2}}{1 + x ^{2}}

= \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

= \frac{3 1 + 3 ^{2}}{1 + 3 ^{2}}

Simplify

= \frac{3 10}{10}

Rewrite using trig identities:

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

cos (arctan (3))

Rewrite using trig identities:

cos (arctan (3)) = \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

Use the following identity:

cos (arctan (x)) = \frac{1 + x ^{2}}{1 + x ^{2}}

= \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

= \frac{1 + 3 ^{2}}{1 + 3 ^{2}}

Simplify

= \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}

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

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

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

31010 \cdot 2 = 60

31010 \cdot 2

Multiply the numbers:

3 \cdot 2 = 6

= 61010

Apply radical rule:

a a = a

1010 = 10

= 6 \cdot 10

Multiply the numbers:

6 \cdot 10 = 60

= 60

= \frac{60}{10 \cdot 10}

Multiply the numbers:

10 \cdot 10 = 100

= \frac{60}{100}

Cancel the common factor:

20

= \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)