What is the value of (cos^4(pi/4))/4 ?

The value of (cos^4(pi/4))/4 is 1/16

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(cos^4(pi/4))/4

\frac{cos ^{4} ( \frac{π}{4} )}{4}

\frac{1}{16}

Decimal Notation

0.0625

Solution steps

\frac{cos ^{4} ( \frac{π}{4} )}{4}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

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

Simplify

\frac{( \frac{2}{2} ) ^{4}}{4} : \frac{1}{16}

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

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

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

Apply exponent rule:

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

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

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

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 4

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2^{2}

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

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

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

Subtract the numbers:

4 - 2 = 2

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

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

Apply the fraction rule:

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

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

2^{2} \cdot 4 = 16

2^{2} \cdot 4

2^{2} = 4

= 4 \cdot 4

Multiply the numbers:

4 \cdot 4 = 16

= 16

= \frac{1}{16}

= \frac{1}{16}

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