What is the value of sin(2arcsin(1/3)) ?

The value of sin(2arcsin(1/3)) is (4sqrt(2))/9

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sin(2arcsin(1/3))

sin (2 arcsin (\frac{1}{3}))

\frac{4 2}{9}

Decimal Notation

0.62853 \dots

Solution steps

sin (2 arcsin (\frac{1}{3}))

Rewrite using trig identities:

2 sin (arcsin (\frac{1}{3})) cos (arcsin (\frac{1}{3}))

sin (2 arcsin (\frac{1}{3}))

Use the Double Angle identity:

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

= 2 sin (arcsin (\frac{1}{3})) cos (arcsin (\frac{1}{3}))

= 2 sin (arcsin (\frac{1}{3})) cos (arcsin (\frac{1}{3}))

Rewrite using trig identities:

sin (arcsin (\frac{1}{3})) = \frac{1}{3}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{3}

Rewrite using trig identities:

cos (arcsin (\frac{1}{3})) = \frac{2 2}{3}

cos (arcsin (\frac{1}{3}))

Rewrite using trig identities:

cos (arcsin (\frac{1}{3})) = 1 - (\frac{1}{3})^{2}

Use the following identity:

cos (arcsin (x)) = 1 - x^{2}

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

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

Simplify

= \frac{2 2}{3}

= 2 \cdot \frac{1}{3} \cdot \frac{2 2}{3}

Simplify

2 \cdot \frac{1}{3} \cdot \frac{2 2}{3} : \frac{4 2}{9}

2 \cdot \frac{1}{3} \cdot \frac{2 2}{3}

Multiply fractions:

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

= \frac{1 \cdot 2 2 \cdot 2}{3 \cdot 3}

Multiply the numbers:

1 \cdot 2 \cdot 2 = 4

= \frac{4 2}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{4 2}{9}

= \frac{4 2}{9}

\frac{3}{2} sec (\frac{π}{3})

cot (- \frac{π}{12})

arcsin (\frac{2}{8})

tan (\frac{37 π}{4}) + sec (\frac{175 π}{4})

sin (arccos (- \frac{2}{3}) + 2 arctan (- \frac{1}{3}))

What is the value of sin(2arcsin(1/3)) ?
The value of sin(2arcsin(1/3)) is (4sqrt(2))/9