What is the general solution for 5sec(x)csc(x)=10csc(x) ?

The general solution for 5sec(x)csc(x)=10csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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5sec(x)csc(x)=10csc(x)

5 sec (x) csc (x) = 10 csc (x)

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (x) csc (x) = 10 csc (x)

Subtract

10 csc (x)

from both sides

5 sec (x) csc (x) - 10 csc (x) = 0

Factor

5 sec (x) csc (x) - 10 csc (x) : 5 csc (x) (sec (x) - 2)

5 sec (x) csc (x) - 10 csc (x)

Rewrite

- 10

2 \cdot 5

= 5 sec (x) csc (x) + 2 \cdot 5 csc (x)

Factor out common term

5 csc (x)

= 5 csc (x) (sec (x) - 2)

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

sec (x) - 2 = 0

Move

2

to the right side

sec (x) - 2 = 0

Add

2

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

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

Combine all the solutions

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

cos (x) = 1 - sin (x)

2 sin (2 θ) = cos (2 θ), 0 \leq θ \leq 2 π

tan (4 x) - 1 = 0

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = 3

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

What is the general solution for 5sec(x)csc(x)=10csc(x) ?
The general solution for 5sec(x)csc(x)=10csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin