What is the value of sin(2arctan(-12/5)) ?

The value of sin(2arctan(-12/5)) is -120/169

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sin(2arctan(-12/5))

sin (2 arctan (- \frac{12}{5}))

- \frac{120}{169}

Decimal Notation

- 0.71005 \dots

Solution steps

sin (2 arctan (- \frac{12}{5}))

Rewrite using trig identities:

- 2 sin (arctan (\frac{12}{5})) cos (arctan (\frac{12}{5}))

sin (2 arctan (- \frac{12}{5}))

Use the Double Angle identity:

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

= 2 sin (arctan (- \frac{12}{5})) cos (arctan (- \frac{12}{5}))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{12}{5}) = - arctan (\frac{12}{5})

= 2 sin (arctan (- \frac{12}{5})) cos (- arctan (\frac{12}{5}))

Use the following property:

cos (- x) = cos (x)

cos (- arctan (\frac{12}{5})) = cos (arctan (\frac{12}{5}))

= 2 sin (arctan (- \frac{12}{5})) cos (arctan (\frac{12}{5}))

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{12}{5}) = - arctan (\frac{12}{5})

= 2 sin (- arctan (\frac{12}{5})) cos (arctan (\frac{12}{5}))

Use the following property:

sin (- x) = - sin (x)

sin (- arctan (\frac{12}{5})) = - sin (arctan (\frac{12}{5}))

= 2 (- sin (arctan (\frac{12}{5}))) cos (arctan (\frac{12}{5}))

Simplify

= - 2 sin (arctan (\frac{12}{5})) cos (arctan (\frac{12}{5}))

= - 2 sin (arctan (\frac{12}{5})) cos (arctan (\frac{12}{5}))

Rewrite using trig identities:

sin (arctan (\frac{12}{5})) = \frac{12}{13}

sin (arctan (\frac{12}{5}))

Rewrite using trig identities:

sin (arctan (\frac{12}{5})) = \frac{( \frac{12}{5} ) 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Use the following identity:

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

= \frac{( \frac{12}{5} ) 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

= \frac{\frac{12}{5} 1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Simplify

= \frac{12}{13}

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{5}{13}

cos (arctan (\frac{12}{5}))

Rewrite using trig identities:

cos (arctan (\frac{12}{5})) = \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Use the following identity:

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

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

= \frac{1 + ( \frac{12}{5} ) ^{2}}{1 + ( \frac{12}{5} ) ^{2}}

Simplify

= \frac{5}{13}

= - 2 \cdot \frac{12}{13} \cdot \frac{5}{13}

Simplify

- 2 \cdot \frac{12}{13} \cdot \frac{5}{13} : - \frac{120}{169}

- 2 \cdot \frac{12}{13} \cdot \frac{5}{13}

Convert element to fraction:

2 = \frac{2}{1}

= \frac{2}{1} \cdot \frac{5}{13} \cdot \frac{12}{13}

Multiply fractions:

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

= - \frac{2 \cdot 12 \cdot 5}{1 \cdot 13 \cdot 13}

Multiply the numbers:

2 \cdot 12 \cdot 5 = 120

= - \frac{120}{1 \cdot 13 \cdot 13}

Multiply the numbers:

1 \cdot 13 \cdot 13 = 169

= - \frac{120}{169}

= - \frac{120}{169}

cos (\frac{6}{10})

tan (arcsec (\frac{5}{3}))

sin (2) (80)

\frac{18}{tan ( 1 7 ^{\circ} )}

5 \cdot cos (4 0^{\circ})

What is the value of sin(2arctan(-12/5)) ?
The value of sin(2arctan(-12/5)) is -120/169