What is the general solution for tan(x)(csc(x)+2)=0 ?

The general solution for tan(x)(csc(x)+2)=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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

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

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

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solving each part separately

tan (x) = 0 or csc (x) + 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

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

Since the equation is undefined for:

πn

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

4 cos^{2} (x) = 5 + 4 sin (x), 0 \leq x \leq 2 π

sin (π + θ) = sin (θ)

0 = 2 cos (x) + sin (x)

2 sin^{3} (x) - sin^{2} (x) - sin (x) = 0

tan (t) + sec (t) = 1

What is the general solution for tan(x)(csc(x)+2)=0 ?
The general solution for tan(x)(csc(x)+2)=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin