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

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

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

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

- 1

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

Rewrite using trig identities:

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

Use the following identity:

cos (arccos (x)) = x

= - \frac{2}{2}

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Multiply fractions:

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

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

Cancel the common factor:

2

= - 2 \frac{1}{2}

Apply radical rule:

a b = a \cdot b

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

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

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

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= - 1

Apply rule

1 = 1

= - 1

= - 1

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