What is the general solution for csc^2(2x-0.6)=16 ?

The general solution for csc^2(2x-0.6)=16 is

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csc^2(2x-0.6)=16

Solution

csc^{2} (2 x - 0.6) = 16

Solution

x = πn + \frac{0.6}{2} + \frac{0.25268 \dots}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{0.25268 \dots}{2}, x = πn + \frac{0.6}{2} - \frac{0.25268 \dots}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{0.25268 \dots}{2}

Degrees

x = 24.42748 \dots^{\circ} + 18 0^{\circ} n, x = 99.94997 \dots^{\circ} + 18 0^{\circ} n, x = 9.94997 \dots^{\circ} + 18 0^{\circ} n, x = 114.42748 \dots^{\circ} + 18 0^{\circ} n

Solution steps

csc^{2} (2 x - 0.6) = 16

Solve by substitution

csc^{2} (2 x - 0.6) = 16

Let:

csc (2 x - 0.6) = u

u^{2} = 16

u^{2} = 16 : u = 4, u = - 4

u^{2} = 16

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 16, u = - 16

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

- 16 = - 4

- 16

16 = 4

16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

= - 4

u = 4, u = - 4

Substitute back

u = csc (2 x - 0.6)

csc (2 x - 0.6) = 4, csc (2 x - 0.6) = - 4

csc (2 x - 0.6) = 4, csc (2 x - 0.6) = - 4

csc (2 x - 0.6) = 4 : x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

csc (2 x - 0.6) = 4

Apply trig inverse properties

csc (2 x - 0.6) = 4

General solutions for

csc (2 x - 0.6) = 4

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

2 x - 0.6 = arccsc (4) + 2 πn, 2 x - 0.6 = π - arccsc (4) + 2 πn

2 x - 0.6 = arccsc (4) + 2 πn, 2 x - 0.6 = π - arccsc (4) + 2 πn

Solve

2 x - 0.6 = arccsc (4) + 2 πn : x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

2 x - 0.6 = arccsc (4) + 2 πn

Move

0.6

to the right side

2 x - 0.6 = arccsc (4) + 2 πn

Add

0.6

to both sides

2 x - 0.6 + 0.6 = arccsc (4) + 2 πn + 0.6

Simplify

2 x = arccsc (4) + 2 πn + 0.6

2 x = arccsc (4) + 2 πn + 0.6

Divide both sides by

2

2 x = arccsc (4) + 2 πn + 0.6

Divide both sides by

2

\frac{2 x}{2} = \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} = \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2} : πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

\frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Group like terms

= \frac{0.6}{2} + \frac{2 πn}{2} + \frac{arccsc ( 4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{0.6}{2} + πn + \frac{arccsc ( 4 )}{2}

Group like terms

= πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

Solve

2 x - 0.6 = π - arccsc (4) + 2 πn : x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

2 x - 0.6 = π - arccsc (4) + 2 πn

Move

0.6

to the right side

2 x - 0.6 = π - arccsc (4) + 2 πn

Add

0.6

to both sides

2 x - 0.6 + 0.6 = π - arccsc (4) + 2 πn + 0.6

Simplify

2 x = π - arccsc (4) + 2 πn + 0.6

2 x = π - arccsc (4) + 2 πn + 0.6

Divide both sides by

2

2 x = π - arccsc (4) + 2 πn + 0.6

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2} : πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

\frac{π}{2} - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Group like terms

= \frac{π}{2} + \frac{0.6}{2} + \frac{2 πn}{2} - \frac{arccsc ( 4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π}{2} + \frac{0.6}{2} + πn - \frac{arccsc ( 4 )}{2}

Group like terms

= πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

csc (2 x - 0.6) = - 4 : x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

csc (2 x - 0.6) = - 4

Apply trig inverse properties

csc (2 x - 0.6) = - 4

General solutions for

csc (2 x - 0.6) = - 4

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

2 x - 0.6 = arccsc (- 4) + 2 πn, 2 x - 0.6 = π + arccsc (4) + 2 πn

2 x - 0.6 = arccsc (- 4) + 2 πn, 2 x - 0.6 = π + arccsc (4) + 2 πn

Solve

2 x - 0.6 = arccsc (- 4) + 2 πn : x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

2 x - 0.6 = arccsc (- 4) + 2 πn

Simplify

arccsc (- 4) + 2 πn : - arccsc (4) + 2 πn

arccsc (- 4) + 2 πn

Use the following property:

arccsc (- x) = - arccsc (x)

arccsc (- 4) = - arccsc (4)

= - arccsc (4) + 2 πn

2 x - 0.6 = - arccsc (4) + 2 πn

Move

0.6

to the right side

2 x - 0.6 = - arccsc (4) + 2 πn

Add

0.6

to both sides

2 x - 0.6 + 0.6 = - arccsc (4) + 2 πn + 0.6

Simplify

2 x = - arccsc (4) + 2 πn + 0.6

2 x = - arccsc (4) + 2 πn + 0.6

Divide both sides by

2

2 x = - arccsc (4) + 2 πn + 0.6

Divide both sides by

2

\frac{2 x}{2} = - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} = - \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2} : πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

- \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Group like terms

= \frac{0.6}{2} + \frac{2 πn}{2} - \frac{arccsc ( 4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{0.6}{2} + πn - \frac{arccsc ( 4 )}{2}

Group like terms

= πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}

Solve

2 x - 0.6 = π + arccsc (4) + 2 πn : x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

2 x - 0.6 = π + arccsc (4) + 2 πn

Move

0.6

to the right side

2 x - 0.6 = π + arccsc (4) + 2 πn

Add

0.6

to both sides

2 x - 0.6 + 0.6 = π + arccsc (4) + 2 πn + 0.6

Simplify

2 x = π + arccsc (4) + 2 πn + 0.6

2 x = π + arccsc (4) + 2 πn + 0.6

Divide both sides by

2

2 x = π + arccsc (4) + 2 πn + 0.6

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} = \frac{π}{2} + \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{2} + \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2} : πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

\frac{π}{2} + \frac{arccsc ( 4 )}{2} + \frac{2 πn}{2} + \frac{0.6}{2}

Group like terms

= \frac{π}{2} + \frac{0.6}{2} + \frac{2 πn}{2} + \frac{arccsc ( 4 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= \frac{π}{2} + \frac{0.6}{2} + πn + \frac{arccsc ( 4 )}{2}

Group like terms

= πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

Combine all the solutions

x = πn + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}, x = πn + \frac{0.6}{2} - \frac{arccsc ( 4 )}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{arccsc ( 4 )}{2}

Show solutions in decimal form

x = πn + \frac{0.6}{2} + \frac{0.25268 \dots}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} - \frac{0.25268 \dots}{2}, x = πn + \frac{0.6}{2} - \frac{0.25268 \dots}{2}, x = πn + \frac{π}{2} + \frac{0.6}{2} + \frac{0.25268 \dots}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for csc^2(2x-0.6)=16 ?
The general solution for csc^2(2x-0.6)=16 is