Is cos(pi/6)= 4/3 cos^3(pi/6) ?

The answer to whether cos(pi/6)= 4/3 cos^3(pi/6) is True

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prove cos(pi/6)= 4/3 cos^3(pi/6)

Solution

prove

cos (\frac{π}{6}) = \frac{4}{3} cos^{3} (\frac{π}{6})

Solution

True

Solution steps

cos (\frac{π}{6}) = \frac{4}{3} cos^{3} (\frac{π}{6})

Manipulating left side

cos (\frac{π}{6})

Simplify

= \frac{3}{2}

Manipulating right side

\frac{4}{3} cos^{3} (\frac{π}{6})

Simplify

cos^{3} (\frac{π}{6}) \frac{4}{3} : \frac{3}{2}

cos^{3} (\frac{π}{6}) \frac{4}{3}

cos^{3} (\frac{π}{6}) = \frac{3 3}{2 ^{3}}

cos^{3} (\frac{π}{6})

Simplify

cos (\frac{π}{6}) : \frac{3}{2}

cos (\frac{π}{6})

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

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{3}{2}

= (\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}}

= \frac{4}{3} \cdot \frac{3 3}{2 ^{3}}

Simplify

\frac{3 3}{2 ^{3}} \cdot \frac{4}{3}

Multiply fractions:

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

= \frac{3 3 \cdot 4}{2 ^{3} \cdot 3}

Cancel the common factor:

3

= \frac{3 \cdot 4}{2 ^{3}}

Factor

4 : 2^{2}

Factor

4 = 2^{2}

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

Cancel

\frac{3 \cdot 2 ^{2}}{2 ^{3}} : \frac{3}{2}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

3 - 2 = 1

= \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2}

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is cos(pi/6)= 4/3 cos^3(pi/6) ?
The answer to whether cos(pi/6)= 4/3 cos^3(pi/6) is True