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

The value of sin(2arcsin((sqrt(2))/2)) is 1

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sin(2arcsin((sqrt(2))/2))

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

1

Solution steps

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

Rewrite using trig identities:

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

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{2}{2}

Rewrite using trig identities:

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

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

Rewrite using trig identities:

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

Use the following identity:

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

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

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

Simplify

= \frac{1}{2}

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

Simplify

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

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

tan (\frac{7}{24})

cos (\frac{3}{π})

arctan (\frac{- 3}{0})

\frac{3 cos ( π )}{sin ( π ) + 6}

arccos (10)

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