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Popular >

solvefor x,sin(t)-csc(x)+1=0

Solution

solve for

x, sin (t) - csc (x) + 1 = 0

Solution

x = arccsc (sin (t) + 1) + 2 πn, x = π + arccsc (- sin (t) - 1) + 2 πn

Solution steps

sin (t) - csc (x) + 1 = 0

Move

sin (t)

to the right side

sin (t) - csc (x) + 1 = 0

Subtract

sin (t)

from both sides

sin (t) - csc (x) + 1 - sin (t) = 0 - sin (t)

Simplify

- csc (x) + 1 = - sin (t)

- csc (x) + 1 = - sin (t)

Move

1

to the right side

- csc (x) + 1 = - sin (t)

Subtract

1

from both sides

- csc (x) + 1 - 1 = - sin (t) - 1

Simplify

- csc (x) = - sin (t) - 1

- csc (x) = - sin (t) - 1

Divide both sides by

- 1

- csc (x) = - sin (t) - 1

Divide both sides by

- 1

\frac{- csc ( x )}{- 1} = - \frac{sin ( t )}{- 1} - \frac{1}{- 1}

Simplify

\frac{- csc ( x )}{- 1} = - \frac{sin ( t )}{- 1} - \frac{1}{- 1}

Simplify

\frac{- csc ( x )}{- 1} : csc (x)

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Apply rule

\frac{a}{1} = a

= csc (x)

Simplify

- \frac{sin ( t )}{- 1} - \frac{1}{- 1} : sin (t) + 1

- \frac{sin ( t )}{- 1} - \frac{1}{- 1}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- sin ( t ) - 1}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- sin ( t ) - 1}{1}

Apply rule

\frac{a}{1} = a

= - (- sin (t) - 1)

Distribute parentheses

= - (- sin (t)) - (- 1)

Apply minus-plus rules

- (- a) = a

= sin (t) + 1

csc (x) = sin (t) + 1

csc (x) = sin (t) + 1

csc (x) = sin (t) + 1

Apply trig inverse properties

csc (x) = sin (t) + 1

General solutions for

csc (x) = sin (t) + 1

csc (x) = a \Rightarrow x = arccsc (a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (sin (t) + 1) + 2 πn, x = π + arccsc (- sin (t) - 1) + 2 πn

x = arccsc (sin (t) + 1) + 2 πn, x = π + arccsc (- sin (t) - 1) + 2 πn

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