What is the general solution for -4sin(x)=4(sin(x)-1) ?

The general solution for -4sin(x)=4(sin(x)-1) is x= pi/6+2pin,x=(5pi)/6+2pin

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-4sin(x)=4(sin(x)-1)

Solution

- 4 sin (x) = 4 (sin (x) - 1)

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

- 4 sin (x) = 4 (sin (x) - 1)

Solve by substitution

- 4 sin (x) = 4 (sin (x) - 1)

Let:

sin (x) = u

- 4 u = 4 (u - 1)

- 4 u = 4 (u - 1) : u = \frac{1}{2}

- 4 u = 4 (u - 1)

Expand

4 (u - 1) : 4 u - 4

4 (u - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = u, c = 1

= 4 u - 4 \cdot 1

Multiply the numbers:

4 \cdot 1 = 4

= 4 u - 4

- 4 u = 4 u - 4

Move

4 u

to the left side

- 4 u = 4 u - 4

Subtract

4 u

from both sides

- 4 u - 4 u = 4 u - 4 - 4 u

Simplify

- 8 u = - 4

- 8 u = - 4

Divide both sides by

- 8

- 8 u = - 4

Divide both sides by

- 8

\frac{- 8 u}{- 8} = \frac{- 4}{- 8}

Simplify

\frac{- 8 u}{- 8} = \frac{- 4}{- 8}

Simplify

\frac{- 8 u}{- 8} : u

\frac{- 8 u}{- 8}

Apply the fraction rule:

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

= \frac{8 u}{8}

Divide the numbers:

\frac{8}{8} = 1

= u

Simplify

\frac{- 4}{- 8} : \frac{1}{2}

\frac{- 4}{- 8}

Apply the fraction rule:

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

= \frac{4}{8}

Cancel the common factor:

4

= \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{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 -4sin(x)=4(sin(x)-1) ?
The general solution for -4sin(x)=4(sin(x)-1) is x= pi/6+2pin,x=(5pi)/6+2pin