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

The value of sin(2arcsin(-(sqrt(3))/2)) is -(sqrt(3))/2

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sin(2arcsin(-(sqrt(3))/2))

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

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

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

Use the following property:

cos (- x) = cos (x)

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

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

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{3}{2}) = - arcsin (\frac{3}{2})

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

Use the following property:

sin (- x) = - sin (x)

sin (- arcsin (\frac{3}{2})) = - sin (arcsin (\frac{3}{2}))

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

Simplify

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

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

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{3}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

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

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

3 \cdot 1 = 3

= - \frac{3}{2}

= - \frac{3}{2}

arccos (\frac{6}{15})

1 - sin^{2} (\frac{5 π}{3})

- sin (- 9 0^{\circ})

cos (arcsin (\frac{8}{17}))

arccos (\frac{6}{11})

What is the value of sin(2arcsin(-(sqrt(3))/2)) ?
The value of sin(2arcsin(-(sqrt(3))/2)) is -(sqrt(3))/2