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

The general solution for (cot(x)-sqrt(3))(csc(x)+2)=0 is x= pi/6+pin,x=(11pi)/6+2pin

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(cot(x)-sqrt(3))(csc(x)+2)=0

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

x = \frac{π}{6} + πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 18 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solving each part separately

cot (x) - 3 = 0 or csc (x) + 2 = 0

cot (x) - 3 = 0 : x = \frac{π}{6} + πn

cot (x) - 3 = 0

Move

3

to the right side

cot (x) - 3 = 0

Add

3

to both sides

cot (x) - 3 + 3 = 0 + 3

Simplify

cot (x) = 3

cot (x) = 3

General solutions for

cot (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

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

csc (x) + 2 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

csc (x) + 2 = 0

Move

2

to the right side

csc (x) + 2 = 0

Subtract

2

from both sides

csc (x) + 2 - 2 = 0 - 2

Simplify

csc (x) = - 2

csc (x) = - 2

General solutions for

csc (x) = - 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Merge Overlapping Intervals

x = \frac{π}{6} + πn, x = \frac{11 π}{6} + 2 πn

sin (x) = \frac{7}{9.4}

cos (x) = \frac{6}{11}

3 cos^{2} (x) + 5 sin (x) - 1 = 0

2 cos^{2} (θ) = 7 cos (θ) - 3

sin (θ) = (\frac{173}{210})

What is the general solution for (cot(x)-sqrt(3))(csc(x)+2)=0 ?
The general solution for (cot(x)-sqrt(3))(csc(x)+2)=0 is x= pi/6+pin,x=(11pi)/6+2pin