What is the general solution for 15sin(x)+7=sin(x) ?

The general solution for 15sin(x)+7=sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin

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Popular Trigonometry >

15sin(x)+7=sin(x)

Solution

15 sin (x) + 7 = sin (x)

Solution

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

15 sin (x) + 7 = sin (x)

Solve by substitution

15 sin (x) + 7 = sin (x)

Let:

sin (x) = u

15 u + 7 = u

15 u + 7 = u : u = - \frac{1}{2}

15 u + 7 = u

Move

7

to the right side

15 u + 7 = u

Subtract

7

from both sides

15 u + 7 - 7 = u - 7

Simplify

15 u = u - 7

15 u = u - 7

Move

u

to the left side

15 u = u - 7

Subtract

u

from both sides

15 u - u = u - 7 - u

Simplify

14 u = - 7

14 u = - 7

Divide both sides by

14

14 u = - 7

Divide both sides by

14

\frac{14 u}{14} = \frac{- 7}{14}

Simplify

\frac{14 u}{14} = \frac{- 7}{14}

Simplify

\frac{14 u}{14} : u

\frac{14 u}{14}

Divide the numbers:

\frac{14}{14} = 1

= u

Simplify

\frac{- 7}{14} : - \frac{1}{2}

\frac{- 7}{14}

Apply the fraction rule:

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

= - \frac{7}{14}

Cancel the common factor:

7

= - \frac{1}{2}

u = - \frac{1}{2}

u = - \frac{1}{2}

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{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 15sin(x)+7=sin(x) ?
The general solution for 15sin(x)+7=sin(x) is x=(7pi)/6+2pin,x=(11pi)/6+2pin