What is the value of cos(2arcsin(1/2)) ?

The value of cos(2arcsin(1/2)) is 1/2

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cos(2arcsin(1/2))

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

\frac{1}{2}

Decimal Notation

0.5

Solution steps

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

Rewrite using trig identities:

1 - 2 sin^{2} (arcsin (\frac{1}{2}))

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

Use the Double Angle identity:

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

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

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

Rewrite using trig identities:

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

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{2}

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{2 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{2 ^{2}}

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

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{2 ^{2}}

Cancel the common factor:

2

= \frac{1}{2}

= 1 - \frac{1}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiply the numbers:

1 \cdot 2 = 2

= 2 - 1

Subtract the numbers:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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