What is the value of 4/(2-cos(pi/6)) ?

The value of 4/(2-cos(pi/6)) is (8(4+sqrt(3)))/(13)

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4/(2-cos(pi/6))

\frac{4}{2 - cos ( \frac{π}{6} )}

\frac{8 ( 4 + 3 )}{13}

Decimal Notation

3.52741 \dots

Solution steps

\frac{4}{2 - cos ( \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}

= \frac{4}{2 - \frac{3}{2}}

Simplify

\frac{4}{2 - \frac{3}{2}} : \frac{8 ( 4 + 3 )}{13}

\frac{4}{2 - \frac{3}{2}}

Join

2 - \frac{3}{2} : \frac{4 - 3}{2}

2 - \frac{3}{2}

Convert element to fraction:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 - 3}{2}

= \frac{4}{\frac{4 - 3}{2}}

Apply the fraction rule:

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

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

Multiply the numbers:

4 \cdot 2 = 8

= \frac{8}{4 - 3}

Rationalize

\frac{8}{4 - 3} : \frac{8 ( 4 + 3 )}{13}

\frac{8}{4 - 3}

Multiply by the conjugate

\frac{4 + 3}{4 + 3}

= \frac{8 ( 4 + 3 )}{( 4 - 3 ) ( 4 + 3 )}

(4 - 3) (4 + 3) = 13

(4 - 3) (4 + 3)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = 4, b = 3

= 4^{2} - (3)^{2}

Simplify

4^{2} - (3)^{2} : 13

4^{2} - (3)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

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

Apply exponent rule:

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

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 16 - 3

Subtract the numbers:

16 - 3 = 13

= 13

= 13

= \frac{8 ( 4 + 3 )}{13}

= \frac{8 ( 4 + 3 )}{13}

= \frac{8 ( 4 + 3 )}{13}

(sec^{2} (4 5^{\circ}) + tan (4 5^{\circ})) cot (3 7^{\circ}) - 2 cos (6 0^{\circ})

6 π cos (3 π^{0})

tan (49.9 6^{\circ})

sin (24 7^{\circ})

tan (7 0^{\circ}) cot (7 0^{\circ})

What is the value of 4/(2-cos(pi/6)) ?
The value of 4/(2-cos(pi/6)) is (8(4+sqrt(3)))/(13)