What is the value of cos(2arcsin(1/4)) ?

The value of cos(2arcsin(1/4)) is 7/8

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cos(2arcsin(1/4))

cos (2 arcsin (\frac{1}{4}))

\frac{7}{8}

Decimal Notation

0.875

Solution steps

cos (2 arcsin (\frac{1}{4}))

Rewrite using trig identities:

1 - 2 sin^{2} (arcsin (\frac{1}{4}))

cos (2 arcsin (\frac{1}{4}))

Use the Double Angle identity:

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

= 1 - 2 sin^{2} (arcsin (\frac{1}{4}))

= 1 - 2 sin^{2} (arcsin (\frac{1}{4}))

Rewrite using trig identities:

sin (arcsin (\frac{1}{4})) = \frac{1}{4}

Use the following identity:

sin (arcsin (x)) = x

= \frac{1}{4}

= 1 - 2 (\frac{1}{4})^{2}

Simplify

1 - 2 (\frac{1}{4})^{2} : \frac{7}{8}

1 - 2 (\frac{1}{4})^{2}

2 (\frac{1}{4})^{2} = \frac{1}{8}

2 (\frac{1}{4})^{2}

(\frac{1}{4})^{2} = \frac{1}{4 ^{2}}

(\frac{1}{4})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{4 ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{4 ^{2}}

= 2 \cdot \frac{1}{4 ^{2}}

Multiply fractions:

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

= \frac{1 \cdot 2}{4 ^{2}}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2}{4 ^{2}}

Factor

4^{2} : 2^{4}

Factor

4 = 2^{2}

= (2^{2})^{2}

Simplify

(2^{2})^{2} : 2^{4}

(2^{2})^{2}

Apply exponent rule:

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

= 2^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= 2^{4}

= 2^{4}

= \frac{2}{2 ^{4}}

Cancel the common factor:

2

= \frac{1}{2 ^{3}}

2^{3} = 8

= \frac{1}{8}

= 1 - \frac{1}{8}

Convert element to fraction:

1 = \frac{1 \cdot 8}{8}

= \frac{1 \cdot 8}{8} - \frac{1}{8}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 8 - 1}{8}

1 \cdot 8 - 1 = 7

1 \cdot 8 - 1

Multiply the numbers:

1 \cdot 8 = 8

= 8 - 1

Subtract the numbers:

8 - 1 = 7

= 7

= \frac{7}{8}

= \frac{7}{8}

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