What is the value of (cot((3pi)/2)-cot(pi/6))/((3pi)/2-pi/6) ?

The value of (cot((3pi)/2)-cot(pi/6))/((3pi)/2-pi/6) is -(3sqrt(3))/(4pi)

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Popular Trigonometry >

(cot((3pi)/2)-cot(pi/6))/((3pi)/2-pi/6)

Solution

\frac{cot ( \frac{3 π}{2} ) - cot ( \frac{π}{6} )}{\frac{3 π}{2} - \frac{π}{6}}

Solution

- \frac{3 3}{4 π}

Decimal

- 0.41349 \dots

Solution steps

\frac{cot ( \frac{3 π}{2} ) - cot ( \frac{π}{6} )}{\frac{3 π}{2} - \frac{π}{6}}

Rewrite using trig identities:

cot (\frac{3 π}{2}) = 0

cot (\frac{3 π}{2})

cot (\frac{3 π}{2}) = cot (\frac{π}{2})

cot (\frac{3 π}{2})

Rewrite

\frac{3 π}{2}

π + \frac{π}{2}

= cot (π + \frac{π}{2})

Apply the periodicity of

cot

cot (x + π) = cot (x)

cot (π + \frac{π}{2}) = cot (\frac{π}{2})

= cot (\frac{π}{2})

= cot (\frac{π}{2})

Use the following trivial identity:

cot (\frac{π}{2}) = 0

cot (\frac{π}{2})

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 0

= 0

Use the following trivial identity:

cot (\frac{π}{6}) = 3

cot (\frac{π}{6})

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

= 3

= \frac{0 - 3}{\frac{3 π}{2} - \frac{π}{6}}

Simplify

\frac{0 - 3}{\frac{3 π}{2} - \frac{π}{6}} : - \frac{3 3}{4 π}

\frac{0 - 3}{\frac{3 π}{2} - \frac{π}{6}}

0 - 3 = - 3

= \frac{- 3}{\frac{3 π}{2} - \frac{π}{6}}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{3}{\frac{3 π}{2} - \frac{π}{6}}

Join

\frac{3 π}{2} - \frac{π}{6} : \frac{4 π}{3}

\frac{3 π}{2} - \frac{π}{6}

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Multiply each factor the greatest number of times it occurs in either

2

6

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= \frac{9 π}{6} - \frac{π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{9 π - π}{6}

Add similar elements:

9 π - π = 8 π

= \frac{8 π}{6}

Cancel the common factor:

2

= \frac{4 π}{3}

= - \frac{3}{\frac{4 π}{3}}

Apply the fraction rule:

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

= - \frac{3 3}{4 π}

= - \frac{3 3}{4 π}

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Frequently Asked Questions (FAQ)

What is the value of (cot((3pi)/2)-cot(pi/6))/((3pi)/2-pi/6) ?
The value of (cot((3pi)/2)-cot(pi/6))/((3pi)/2-pi/6) is -(3sqrt(3))/(4pi)