What is the value of (1+cos(pi/6))/(sin^3(pi/6)) ?

The value of (1+cos(pi/6))/(sin^3(pi/6)) is 4(2+sqrt(3))

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(1+cos(pi/6))/(sin^3(pi/6))

\frac{1 + cos ( \frac{π}{6} )}{sin ^{3} ( \frac{π}{6} )}

4 (2 + 3)

Decimal Notation

14.92820 \dots

Solution steps

\frac{1 + cos ( \frac{π}{6} )}{sin ^{3} ( \frac{π}{6} )}

Use the following trivial identity:

cos (\frac{π}{6}) = \frac{3}{2}

cos (\frac{π}{6})

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}

Use the following trivial identity:

sin (\frac{π}{6}) = \frac{1}{2}

sin (\frac{π}{6})

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

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

Simplify

\frac{1 + \frac{3}{2}}{( \frac{1}{2} ) ^{3}} : 4 (2 + 3)

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{3} = 1

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

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

Join

1 + \frac{3}{2} : \frac{2 + 3}{2}

1 + \frac{3}{2}

Convert element to fraction:

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

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

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 + 3}{2}

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

Divide fractions:

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

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

Refine

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

Cancel the common factor:

2

= 2^{2} (2 + 3)

2^{2} = 4

= 4 (2 + 3)

= 4 (2 + 3)

sin^{2} (\frac{π}{11}) + cos^{2} (\frac{π}{11})

tan (8.2)

arcsin (\frac{3}{12})

25 cos (2 5^{\circ})

\frac{tan ( 0 )}{sin ( 0 )}

What is the value of (1+cos(pi/6))/(sin^3(pi/6)) ?
The value of (1+cos(pi/6))/(sin^3(pi/6)) is 4(2+sqrt(3))