What is the general solution for sqrt(3)cot(pi/2 x)=-3 ?

The general solution for sqrt(3)cot(pi/2 x)=-3 is x= 5/3+2n

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sqrt(3)cot(pi/2 x)=-3

Solution

3 cot (\frac{π}{2} x) = - 3

Solution

x = \frac{5}{3} + 2 n

Degrees

x = 95.49296 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

3 cot (\frac{π}{2} x) = - 3

Divide both sides by

3

3 cot (\frac{π}{2} x) = - 3

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 cot ( \frac{π}{2} x )}{3} : cot (\frac{π}{2} x)

\frac{3 cot ( \frac{π}{2} x )}{3}

Cancel the common factor:

3

= cot (\frac{π}{2} x)

Simplify

\frac{- 3}{3} : - 3

\frac{- 3}{3}

Apply the fraction rule:

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

= - \frac{3}{3}

Apply radical rule:

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

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

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

Apply exponent rule:

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

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

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

= - 3

cot (\frac{π}{2} x) = - 3

cot (\frac{π}{2} x) = - 3

cot (\frac{π}{2} x) = - 3

General solutions for

cot (\frac{π}{2} x) = - 3

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

\frac{π}{2} x = \frac{5 π}{6} + πn

\frac{π}{2} x = \frac{5 π}{6} + πn

Solve

\frac{π}{2} x = \frac{5 π}{6} + πn : x = \frac{5}{3} + 2 n

\frac{π}{2} x = \frac{5 π}{6} + πn

Multiply both sides by

2

\frac{π}{2} x = \frac{5 π}{6} + πn

Multiply both sides by

2

2 \cdot \frac{π}{2} x = 2 \cdot \frac{5 π}{6} + 2 πn

Simplify

2 \cdot \frac{π}{2} x = 2 \cdot \frac{5 π}{6} + 2 πn

Simplify

2 \cdot \frac{π}{2} x : π x

2 \cdot \frac{π}{2} x

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

Simplify

2 \cdot \frac{5 π}{6} + 2 πn : \frac{5 π}{3} + 2 πn

2 \cdot \frac{5 π}{6} + 2 πn

2 \cdot \frac{5 π}{6} = \frac{5 π}{3}

2 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 2}{6}

Multiply the numbers:

5 \cdot 2 = 10

= \frac{10 π}{6}

Cancel the common factor:

2

= \frac{5 π}{3}

= \frac{5 π}{3} + 2 πn

π x = \frac{5 π}{3} + 2 πn

π x = \frac{5 π}{3} + 2 πn

π x = \frac{5 π}{3} + 2 πn

Divide both sides by

π

π x = \frac{5 π}{3} + 2 πn

Divide both sides by

π

\frac{π x}{π} = \frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} = \frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π} : \frac{5}{3} + 2 n

\frac{\frac{5 π}{3}}{π} + \frac{2 πn}{π}

\frac{\frac{5 π}{3}}{π} = \frac{5}{3}

\frac{\frac{5 π}{3}}{π}

Apply the fraction rule:

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

= \frac{5 π}{3 π}

Cancel the common factor:

π

= \frac{5}{3}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

x = \frac{5}{3} + 2 n

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)cot(pi/2 x)=-3 ?
The general solution for sqrt(3)cot(pi/2 x)=-3 is x= 5/3+2n