What is the general solution for 8sin(x)=2+4/(csc(x)) ?

The general solution for 8sin(x)=2+4/(csc(x)) is x= pi/6+2pin,x=(5pi)/6+2pin

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8sin(x)=2+4/(csc(x))

Solution

8 sin (x) = 2 + \frac{4}{csc ( x )}

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

8 sin (x) = 2 + \frac{4}{csc ( x )}

Subtract

2 + \frac{4}{csc ( x )}

from both sides

8 sin (x) - 2 - \frac{4}{csc ( x )} = 0

Simplify

8 sin (x) - 2 - \frac{4}{csc ( x )} : \frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )}

8 sin (x) - 2 - \frac{4}{csc ( x )}

Convert element to fraction:

8 sin (x) = \frac{8 sin ( x ) csc ( x )}{csc ( x )}, 2 = \frac{2 csc ( x )}{csc ( x )}

= \frac{8 sin ( x ) csc ( x )}{csc ( x )} - \frac{2 csc ( x )}{csc ( x )} - \frac{4}{csc ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )}

\frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

8 sin (x) csc (x) - 2 csc (x) - 4 = 0

Rewrite using trig identities

8 sin (x) csc (x) - 2 csc (x) - 4

csc (x) sin (x) = 1

csc (x) sin (x)

Express with sin, cos

csc (x) sin (x)

Use the basic trigonometric identity:

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

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

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

\frac{1}{sin ( x )} sin (x)

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 1

= 1

= - 4 - 2 csc (x) + 8 \cdot 1

- 4 - 2 csc (x) + 8 \cdot 1 = 0

Multiply the numbers:

8 \cdot 1 = 8

- 4 - 2 csc (x) + 8 = 0

Group like terms

- 2 csc (x) - 4 + 8 = 0

Add/Subtract the numbers:

- 4 + 8 = 4

- 2 csc (x) + 4 = 0

Move

4

to the right side

- 2 csc (x) + 4 = 0

Subtract

4

from both sides

- 2 csc (x) + 4 - 4 = 0 - 4

Simplify

- 2 csc (x) = - 4

- 2 csc (x) = - 4

Divide both sides by

- 2

- 2 csc (x) = - 4

Divide both sides by

- 2

\frac{- 2 csc ( x )}{- 2} = \frac{- 4}{- 2}

Simplify

\frac{- 2 csc ( x )}{- 2} = \frac{- 4}{- 2}

Simplify

\frac{- 2 csc ( x )}{- 2} : csc (x)

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

= csc (x)

Simplify

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Apply the fraction rule:

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

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

csc (x) = 2

csc (x) = 2

csc (x) = 2

General solutions for

csc (x) = 2

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 8sin(x)=2+4/(csc(x)) ?
The general solution for 8sin(x)=2+4/(csc(x)) is x= pi/6+2pin,x=(5pi)/6+2pin