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

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

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

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

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

sin (arccos (x)) = 1 - x^{2}

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

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

Simplify

= \frac{1}{2}

Rewrite using trig identities:

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

Use the following identity:

cos (arccos (x)) = x

= - \frac{3}{2}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Multiply:

1 \cdot 3 = 3

= - \frac{3}{2}

= - \frac{3}{2}

- 2 sin (2 (0))

cot (33.6 9^{\circ})

arctan (\frac{8}{0})

cot (\frac{3 π}{5})

arccos (\frac{12}{129 \cdot 29})

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