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

The general solution for sec(θ)csc(θ)+sqrt(2)csc(θ)=0 is θ=(3pi)/4+2pin,θ=(5pi)/4+2pin

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sec(θ)csc(θ)+sqrt(2)csc(θ)=0

sec (θ) csc (θ) + 2 csc (θ) = 0

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Degrees

θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

sec (θ) csc (θ) + 2 csc (θ) = 0

Factor

sec (θ) csc (θ) + 2 csc (θ) : csc (θ) (sec (θ) + 2)

sec (θ) csc (θ) + 2 csc (θ)

Factor out common term

csc (θ)

= csc (θ) (sec (θ) + 2)

csc (θ) (sec (θ) + 2) = 0

Solving each part separately

csc (θ) = 0 or sec (θ) + 2 = 0

csc (θ) = 0 :

No Solution

csc (θ) = 0

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

No Solution

sec (θ) + 2 = 0 : θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

sec (θ) + 2 = 0

Move

2

to the right side

sec (θ) + 2 = 0

Subtract

2

from both sides

sec (θ) + 2 - 2 = 0 - 2

Simplify

sec (θ) = - 2

sec (θ) = - 2

General solutions for

sec (θ) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Combine all the solutions

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

2 sin^{2} (θ) - sin (θ) - 1 = 0, 0 \leq θ \leq 2 π

sin (3 θ) = \frac{2}{2}

(tan^{2} (θ) - 9) (2 cos (θ) + 1) = 0

2 sin (x) + cos (x) = 1

sin (θ) cos (θ) = - \frac{1}{2}

What is the general solution for sec(θ)csc(θ)+sqrt(2)csc(θ)=0 ?
The general solution for sec(θ)csc(θ)+sqrt(2)csc(θ)=0 is θ=(3pi)/4+2pin,θ=(5pi)/4+2pin