What is the general solution for sec(x)*cot(x)-2cot(x)+sec(x)=2 ?

The general solution for sec(x)*cot(x)-2cot(x)+sec(x)=2 is x= pi/3+2pin,x=(5pi)/3+2pin,x=(3pi)/4+pin

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sec(x)*cot(x)-2cot(x)+sec(x)=2

Solution

sec (x) \cdot cot (x) - 2 cot (x) + sec (x) = 2

Solution

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

2

from both sides

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

Factor

sec (x) cot (x) - 2 cot (x) + sec (x) - 2 : (sec (x) - 2) (cot (x) + 1)

sec (x) cot (x) - 2 cot (x) + sec (x) - 2

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

Factor out

cot (x)

from

cot (x) sec (x) - 2 cot (x) : cot (x) (sec (x) - 2)

cot (x) sec (x) - 2 cot (x)

Factor out common term

cot (x)

= cot (x) (sec (x) - 2)

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

Factor out common term

sec (x) - 2

= (sec (x) - 2) (cot (x) + 1)

(sec (x) - 2) (cot (x) + 1) = 0

Solving each part separately

sec (x) - 2 = 0 or cot (x) + 1 = 0

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

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

cot (x) + 1 - 1 = 0 - 1

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

cot (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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

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

Popular Examples

3sin^2(x)-10sin(x)-8=0

3 sin^{2} (x) - 10 sin (x) - 8 = 0

2*9.8*L(1-cos(θ))=0.018

2 \cdot 9.8 \cdot L (1 - cos (θ)) = 0.018

cos(7x)=sin(4x+2)

cos (7 x) = sin (4 x + 2)

12cos(β)-5sin(β)=4.7

12 cos (β) - 5 sin (β) = 4.7

cos(x)=(sqrt(8))/3

cos (x) = \frac{8}{3}

Frequently Asked Questions (FAQ)

What is the general solution for sec(x)*cot(x)-2cot(x)+sec(x)=2 ?
The general solution for sec(x)*cot(x)-2cot(x)+sec(x)=2 is x= pi/3+2pin,x=(5pi)/3+2pin,x=(3pi)/4+pin