What is the general solution for -3cot((3x)/2)csc((3x)/2)=0 ?

The general solution for -3cot((3x)/2)csc((3x)/2)=0 is x= pi/3+(2pin)/3

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-3cot((3x)/2)csc((3x)/2)=0

- 3 cot (\frac{3 x}{2}) csc (\frac{3 x}{2}) = 0

x = \frac{π}{3} + \frac{2 πn}{3}

Degrees

x = 6 0^{\circ} + 12 0^{\circ} n

Solution steps

- 3 cot (\frac{3 x}{2}) csc (\frac{3 x}{2}) = 0

Solving each part separately

cot (\frac{3 x}{2}) = 0 or csc (\frac{3 x}{2}) = 0

cot (\frac{3 x}{2}) = 0 : x = \frac{π}{3} + \frac{2 πn}{3}

cot (\frac{3 x}{2}) = 0

General solutions for

cot (\frac{3 x}{2}) = 0

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{3 x}{2} = \frac{π}{2} + πn

\frac{3 x}{2} = \frac{π}{2} + πn

Solve

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

\frac{3 x}{2} = \frac{π}{2} + πn

Multiply both sides by

2

\frac{3 x}{2} = \frac{π}{2} + πn

Multiply both sides by

2

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

Simplify

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

Simplify

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

\frac{2 \cdot 3 x}{2}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6 x}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3 x

Simplify

2 \cdot \frac{π}{2} + 2 πn : π + 2 πn

2 \cdot \frac{π}{2} + 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= π + 2 πn

3 x = π + 2 πn

3 x = π + 2 πn

3 x = π + 2 πn

Divide both sides by

3

3 x = π + 2 πn

Divide both sides by

3

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

Simplify

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

x = \frac{π}{3} + \frac{2 πn}{3}

csc (\frac{3 x}{2}) = 0 :

No Solution

csc (\frac{3 x}{2}) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

Combine all the solutions

x = \frac{π}{3} + \frac{2 πn}{3}

sec (θ) = \frac{17}{8}

3 sin^{2} (x) = 1

3 sin (θ) + 5 = - 2 sin (θ)

2 - 4 sin (θ) = 0

2 cos (t) = 1, - π \leq t \leq π

What is the general solution for -3cot((3x)/2)csc((3x)/2)=0 ?
The general solution for -3cot((3x)/2)csc((3x)/2)=0 is x= pi/3+(2pin)/3