What is the general solution for (5sin(x)+6)/(sin(x))=17 ?

The general solution for (5sin(x)+6)/(sin(x))=17 is x= pi/6+2pin,x=(5pi)/6+2pin

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(5sin(x)+6)/(sin(x))=17

Solution

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

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

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

Solve by substitution

\frac{5 sin ( x ) + 6}{sin ( x )} = 17

Let:

sin (x) = u

\frac{5 u + 6}{u} = 17

\frac{5 u + 6}{u} = 17 : u = \frac{1}{2}

\frac{5 u + 6}{u} = 17

Multiply both sides by

u

\frac{5 u + 6}{u} = 17

Multiply both sides by

u

\frac{5 u + 6}{u} u = 17 u

Simplify

5 u + 6 = 17 u

5 u + 6 = 17 u

Move

6

to the right side

5 u + 6 = 17 u

Subtract

6

from both sides

5 u + 6 - 6 = 17 u - 6

Simplify

5 u = 17 u - 6

5 u = 17 u - 6

Move

17 u

to the left side

5 u = 17 u - 6

Subtract

17 u

from both sides

5 u - 17 u = 17 u - 6 - 17 u

Simplify

- 12 u = - 6

- 12 u = - 6

Divide both sides by

- 12

- 12 u = - 6

Divide both sides by

- 12

\frac{- 12 u}{- 12} = \frac{- 6}{- 12}

Simplify

\frac{- 12 u}{- 12} = \frac{- 6}{- 12}

Simplify

\frac{- 12 u}{- 12} : u

\frac{- 12 u}{- 12}

Apply the fraction rule:

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

= \frac{12 u}{12}

Divide the numbers:

\frac{12}{12} = 1

= u

Simplify

\frac{- 6}{- 12} : \frac{1}{2}

\frac{- 6}{- 12}

Apply the fraction rule:

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

= \frac{6}{12}

Cancel the common factor:

6

= \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

u = \frac{1}{2}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{5 u + 6}{u}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \frac{1}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{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

Combine all the solutions

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 (5sin(x)+6)/(sin(x))=17 ?
The general solution for (5sin(x)+6)/(sin(x))=17 is x= pi/6+2pin,x=(5pi)/6+2pin