What is the value of sin(2arcsin(5/13)) ?

The value of sin(2arcsin(5/13)) is 120/169

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sin(2arcsin(5/13))

sin (2 arcsin (\frac{5}{13}))

\frac{120}{169}

Decimal Notation

0.71005 \dots

Solution steps

sin (2 arcsin (\frac{5}{13}))

Rewrite using trig identities:

2 sin (arcsin (\frac{5}{13})) cos (arcsin (\frac{5}{13}))

sin (2 arcsin (\frac{5}{13}))

Use the Double Angle identity:

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

= 2 sin (arcsin (\frac{5}{13})) cos (arcsin (\frac{5}{13}))

= 2 sin (arcsin (\frac{5}{13})) cos (arcsin (\frac{5}{13}))

Rewrite using trig identities:

sin (arcsin (\frac{5}{13})) = \frac{5}{13}

Use the following identity:

sin (arcsin (x)) = x

= \frac{5}{13}

Rewrite using trig identities:

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

cos (arcsin (\frac{5}{13}))

Rewrite using trig identities:

cos (arcsin (\frac{5}{13})) = 1 - (\frac{5}{13})^{2}

Use the following identity:

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

= 1 - (\frac{5}{13})^{2}

= 1 - (\frac{5}{13})^{2}

Simplify

= \frac{12}{13}

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

Simplify

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

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

Multiply the numbers:

2 \cdot 5 \cdot 12 = 120

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

Multiply the numbers:

1 \cdot 13 \cdot 13 = 169

= \frac{120}{169}

= \frac{120}{169}

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What is the value of sin(2arcsin(5/13)) ?
The value of sin(2arcsin(5/13)) is 120/169