What is the general solution for sin^3(x)-6sin^2(x)-sin(x)+6=0 ?

The general solution for sin^3(x)-6sin^2(x)-sin(x)+6=0 is x=(3pi)/2+2pin,x= pi/2+2pin

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Popular >

sin^3(x)-6sin^2(x)-sin(x)+6=0

Solution

sin^{3} (x) - 6 sin^{2} (x) - sin (x) + 6 = 0

Solution

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{3} (x) - 6 sin^{2} (x) - sin (x) + 6 = 0

Solve by substitution

sin^{3} (x) - 6 sin^{2} (x) - sin (x) + 6 = 0

Let:

sin (x) = u

u^{3} - 6 u^{2} - u + 6 = 0

u^{3} - 6 u^{2} - u + 6 = 0 : u = 6, u = - 1, u = 1

u^{3} - 6 u^{2} - u + 6 = 0

Factor

u^{3} - 6 u^{2} - u + 6 : (u - 6) (u + 1) (u - 1)

u^{3} - 6 u^{2} - u + 6

= (u^{3} - 6 u^{2}) + (- u + 6)

Factor out

- 1

from

- u + 6 : - (u - 6)

- u + 6

Factor out common term

- 1

= - (u - 6)

Factor out

u^{2}

from

u^{3} - 6 u^{2} : u^{2} (u - 6)

u^{3} - 6 u^{2}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

u^{3} = u u^{2}

= u u^{2} - 6 u^{2}

Factor out common term

u^{2}

= u^{2} (u - 6)

= - (u - 6) + u^{2} (u - 6)

Factor out common term

u - 6

= (u - 6) (u^{2} - 1)

Factor

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

Rewrite

1

1^{2}

= u^{2} - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u - 6) (u + 1) (u - 1)

(u - 6) (u + 1) (u - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u - 6 = 0 or u + 1 = 0 or u - 1 = 0

Solve

u - 6 = 0 : u = 6

u - 6 = 0

Move

6

to the right side

u - 6 = 0

Add

6

to both sides

u - 6 + 6 = 0 + 6

Simplify

u = 6

u = 6

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

The solutions are

u = 6, u = - 1, u = 1

Substitute back

u = sin (x)

sin (x) = 6, sin (x) = - 1, sin (x) = 1

sin (x) = 6, sin (x) = - 1, sin (x) = 1

sin (x) = 6 :

No Solution

sin (x) = 6

- 1 \leq sin (x) \leq 1

No Solution

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

sin (x) = - 1

General solutions for

sin (x) = - 1

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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Popular Examples

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tan (θ) = \frac{8}{13.5}

15sin(3x)=0

15 sin (3 x) = 0

sin(3θ)=-1,0<= θ<= 360

sin (3 θ) = - 1, 0^{\circ} \leq θ \leq 36 0^{\circ}

3csc^2(x)+cot^2(x)-4=0

3 csc^{2} (x) + cot^{2} (x) - 4 = 0

solvefor θ,x=2sec(θ)solve for

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Frequently Asked Questions (FAQ)

What is the general solution for sin^3(x)-6sin^2(x)-sin(x)+6=0 ?
The general solution for sin^3(x)-6sin^2(x)-sin(x)+6=0 is x=(3pi)/2+2pin,x= pi/2+2pin