What is the value of sin(2arccos(-1/2)) ?

The value of sin(2arccos(-1/2)) is -(sqrt(3))/2

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sin(2arccos(-1/2))

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

- \frac{3}{2}

Decimal Notation

- 0.86602 \dots

Solution steps

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

Rewrite using trig identities:

2 sin (arccos (- \frac{1}{2})) cos (arccos (- \frac{1}{2}))

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{3}{2}

Rewrite using trig identities:

cos (arccos (- \frac{1}{2})) = - \frac{1}{2}

Use the following identity:

cos (arccos (x)) = x

= - \frac{1}{2}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

= - 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}

cos (\frac{4 π}{8})

cos (\frac{20}{25})

cos (2 arctan (\frac{15}{8}))

sin^{2} (\frac{7 π}{8})

\frac{1}{sin ( 7 5 ^{\circ} )}

What is the value of sin(2arccos(-1/2)) ?
The value of sin(2arccos(-1/2)) is -(sqrt(3))/2