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} \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}

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