What is the value of cos^3(30) ?

The value of cos^3(30) is (3sqrt(3))/8

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cos^3(30)

cos^{3} (3 0^{\circ})

\frac{3 3}{8}

Decimal

0.64951 \dots

Solution steps

cos^{3} (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= (\frac{3}{2})^{3}

Simplify

(\frac{3}{2})^{3} : \frac{3 3}{8}

(\frac{3}{2})^{3}

Apply exponent rule:

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

= \frac{( 3 ) ^{3}}{2 ^{3}}

(3)^{3} : 3^{\frac{3}{2}}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 3

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

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

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

3^{\frac{3}{2}} = 33

3^{\frac{3}{2}}

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

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

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

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

Refine

= 33

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

2^{3} = 8

= \frac{3 3}{8}

= \frac{3 3}{8}