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}

Decimal Notation

0.92307 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

= 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}

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{2 13}{13}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \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}

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

Multiply fractions:

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

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

213 \cdot 313 \cdot 2 = 156

213 \cdot 313 \cdot 2

Multiply the numbers:

2 \cdot 3 \cdot 2 = 12

= 121313

Apply radical rule:

a a = a

1313 = 13

= 12 \cdot 13

Multiply the numbers:

12 \cdot 13 = 156

= 156

= \frac{156}{13 \cdot 13}

Multiply the numbers:

13 \cdot 13 = 169

= \frac{156}{169}

Cancel the common factor:

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