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

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

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tan(θ)csc(θ)+2tan(θ)=0

tan (θ) csc (θ) + 2 tan (θ) = 0

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

Degrees

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

Solution steps

tan (θ) csc (θ) + 2 tan (θ) = 0

Factor

tan (θ) csc (θ) + 2 tan (θ) : tan (θ) (csc (θ) + 2)

tan (θ) csc (θ) + 2 tan (θ)

Factor out common term

tan (θ)

= tan (θ) (csc (θ) + 2)

tan (θ) (csc (θ) + 2) = 0

Solving each part separately

tan (θ) = 0 or csc (θ) + 2 = 0

tan (θ) = 0 : θ = πn

tan (θ) = 0

General solutions for

tan (θ) = 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}

θ = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

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

csc (θ) + 2 = 0

Move

2

to the right side

csc (θ) + 2 = 0

Subtract

2

from both sides

csc (θ) + 2 - 2 = 0 - 2

Simplify

csc (θ) = - 2

csc (θ) = - 2

General solutions for

csc (θ) = - 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

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

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

Combine all the solutions

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

Since the equation is undefined for:

πn

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

250 = 88 sin (θ) + 242 cos (θ)

4 + 2 csc (\frac{θ}{2}) = 8

2 cos (3 x) = \frac{1}{2}, x = \frac{17 π}{4}

cos (θ) = sin (3 6^{\circ})

2 sec^{2} (x) = tan^{2} (x) + 2 sec (x) + 4

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