What is the value of sin(2arctan(-8/15)) ?

The value of sin(2arctan(-8/15)) is -240/289

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sin(2arctan(-8/15))

sin (2 arctan (- \frac{8}{15}))

- \frac{240}{289}

Decimal

- 0.83044 \dots

Solution steps

sin (2 arctan (- \frac{8}{15}))

Rewrite using trig identities:

- 2 sin (arctan (\frac{8}{15})) cos (arctan (\frac{8}{15}))

sin (2 arctan (- \frac{8}{15}))

Use the Double Angle identity:

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

= 2 sin (arctan (- \frac{8}{15})) cos (arctan (- \frac{8}{15}))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{8}{15}) = - arctan (\frac{8}{15})

= 2 sin (arctan (- \frac{8}{15})) cos (- arctan (\frac{8}{15}))

Use the following property:

cos (- x) = cos (x)

cos (- arctan (\frac{8}{15})) = cos (arctan (\frac{8}{15}))

= 2 sin (arctan (- \frac{8}{15})) cos (arctan (\frac{8}{15}))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{8}{15}) = - arctan (\frac{8}{15})

= 2 sin (- arctan (\frac{8}{15})) cos (arctan (\frac{8}{15}))

Use the following property:

sin (- x) = - sin (x)

sin (- arctan (\frac{8}{15})) = - sin (arctan (\frac{8}{15}))

= 2 (- sin (arctan (\frac{8}{15}))) cos (arctan (\frac{8}{15}))

Simplify

= - 2 sin (arctan (\frac{8}{15})) cos (arctan (\frac{8}{15}))

= - 2 sin (arctan (\frac{8}{15})) cos (arctan (\frac{8}{15}))

Rewrite using trig identities:

sin (arctan (\frac{8}{15})) = \frac{8}{17}

sin (arctan (\frac{8}{15}))

Rewrite using trig identities:

sin (arctan (\frac{8}{15})) = \frac{( \frac{8}{15} ) 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{8}{15} ) 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

= \frac{\frac{8}{15} 1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Simplify

= \frac{8}{17}

Rewrite using trig identities:

cos (arctan (\frac{8}{15})) = \frac{15}{17}

cos (arctan (\frac{8}{15}))

Rewrite using trig identities:

cos (arctan (\frac{8}{15})) = \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

= \frac{1 + ( \frac{8}{15} ) ^{2}}{1 + ( \frac{8}{15} ) ^{2}}

Simplify

= \frac{15}{17}

= - 2 \cdot \frac{8}{17} \cdot \frac{15}{17}

Simplify

- 2 \cdot \frac{8}{17} \cdot \frac{15}{17} : - \frac{240}{289}

- 2 \cdot \frac{8}{17} \cdot \frac{15}{17}

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1} \cdot \frac{8}{17} \cdot \frac{15}{17}

Multiply fractions:

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

= - \frac{2 \cdot 8 \cdot 15}{1 \cdot 17 \cdot 17}

Multiply the numbers:

2 \cdot 8 \cdot 15 = 240

= - \frac{240}{1 \cdot 17 \cdot 17}

Multiply the numbers:

1 \cdot 17 \cdot 17 = 289

= - \frac{240}{289}

= - \frac{240}{289}

What is the value of sin(2arctan(-8/15)) ?
The value of sin(2arctan(-8/15)) is -240/289