What is the value of sin(2arctan(4)) ?

The value of sin(2arctan(4)) is 8/17

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sin(2arctan(4))

sin (2 arctan (4))

\frac{8}{17}

Decimal Notation

0.47058 \dots

Solution steps

sin (2 arctan (4))

Rewrite using trig identities:

2 sin (arctan (4)) cos (arctan (4))

sin (2 arctan (4))

Use the Double Angle identity:

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

= 2 sin (arctan (4)) cos (arctan (4))

= 2 sin (arctan (4)) cos (arctan (4))

Rewrite using trig identities:

sin (arctan (4)) = \frac{4 17}{17}

sin (arctan (4))

Rewrite using trig identities:

sin (arctan (4)) = \frac{4 1 + 4 ^{2}}{1 + 4 ^{2}}

Use the following identity:

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

= \frac{4 1 + 4 ^{2}}{1 + 4 ^{2}}

= \frac{4 1 + 4 ^{2}}{1 + 4 ^{2}}

Simplify

= \frac{4 17}{17}

Rewrite using trig identities:

cos (arctan (4)) = \frac{17}{17}

cos (arctan (4))

Rewrite using trig identities:

cos (arctan (4)) = \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

Use the following identity:

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

= \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

= \frac{1 + 4 ^{2}}{1 + 4 ^{2}}

Simplify

= \frac{17}{17}

= 2 \cdot \frac{4 17}{17} \cdot \frac{17}{17}

Simplify

2 \cdot \frac{4 17}{17} \cdot \frac{17}{17} : \frac{8}{17}

2 \cdot \frac{4 17}{17} \cdot \frac{17}{17}

Multiply fractions:

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

= \frac{4 17 17 \cdot 2}{17 \cdot 17}

41717 \cdot 2 = 136

41717 \cdot 2

Multiply the numbers:

4 \cdot 2 = 8

= 81717

Apply radical rule:

a a = a

1717 = 17

= 8 \cdot 17

Multiply the numbers:

8 \cdot 17 = 136

= 136

= \frac{136}{17 \cdot 17}

Multiply the numbers:

17 \cdot 17 = 289

= \frac{136}{289}

Cancel the common factor:

17

= \frac{8}{17}

= \frac{8}{17}

sec (- \frac{7 π}{2})

\frac{10}{sin ( 4 5 ^{\circ} )}

9 sin (7 5^{\circ})

sin (arcsin (- 1))

30 \cdot sin (4 5^{\circ})