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

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

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sec(x)csc(x)+sqrt(2)csc(x)=0

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

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

Degrees

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

Solution steps

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

Factor

sec (x) csc (x) + 2 csc (x) : csc (x) (sec (x) + 2)

sec (x) csc (x) + 2 csc (x)

Factor out common term

csc (x)

= csc (x) (sec (x) + 2)

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

Solving each part separately

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

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

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

sec (x) + 2 = 0

Move

2

to the right side

sec (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

sec (x) = - 2

sec (x) = - 2

General solutions for

sec (x) = - 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}

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

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

Combine all the solutions

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

sin (x) + 3 cos (x) = 1

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

\frac{1}{2} = cos (x)

cos (θ) = \frac{5}{13}

8 cos^{2} (x) + 4 cos (x) - 4 = 0

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