What is the general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 ?

The general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 is x=(3pi)/4+pin

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Popular >

2tan(x)csc(x)+2csc(x)+tan(x)=-1

Solution

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

Solution

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

Subtract

- 1

from both sides

2 tan (x) csc (x) + 2 csc (x) + tan (x) + 1 = 0

Factor

2 tan (x) csc (x) + 2 csc (x) + tan (x) + 1 : (tan (x) + 1) (2 csc (x) + 1)

2 tan (x) csc (x) + 2 csc (x) + tan (x) + 1

= (2 tan (x) csc (x) + 2 csc (x)) + (tan (x) + 1)

Factor out

2 csc (x)

from

2 tan (x) csc (x) + 2 csc (x) : 2 csc (x) (tan (x) + 1)

2 tan (x) csc (x) + 2 csc (x)

Factor out common term

2 csc (x)

= 2 csc (x) (tan (x) + 1)

= (tan (x) + 1) + 2 csc (x) (tan (x) + 1)

Factor out common term

tan (x) + 1

= (tan (x) + 1) (2 csc (x) + 1)

(tan (x) + 1) (2 csc (x) + 1) = 0

Solving each part separately

tan (x) + 1 = 0 or 2 csc (x) + 1 = 0

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

tan (x) + 1 = 0

Move

1

to the right side

tan (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

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}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

2 csc (x) + 1 = 0 :

No Solution

2 csc (x) + 1 = 0

Move

1

to the right side

2 csc (x) + 1 = 0

Subtract

1

from both sides

2 csc (x) + 1 - 1 = 0 - 1

Simplify

2 csc (x) = - 1

2 csc (x) = - 1

Divide both sides by

2

2 csc (x) = - 1

Divide both sides by

2

\frac{2 csc ( x )}{2} = \frac{- 1}{2}

Simplify

csc (x) = - \frac{1}{2}

csc (x) = - \frac{1}{2}

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

No Solution

Combine all the solutions

x = \frac{3 π}{4} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 ?
The general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 is x=(3pi)/4+pin