What is the general solution for sin^3(x)=sin^3(x)cos(x) ?

The general solution for sin^3(x)=sin^3(x)cos(x) is x=2pin,x=pi+2pin

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sin^3(x)=sin^3(x)cos(x)

Solution

sin^{3} (x) = sin^{3} (x) cos (x)

Solution

x = 2 πn, x = π + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{3} (x) = sin^{3} (x) cos (x)

Subtract

sin^{3} (x) cos (x)

from both sides

sin^{3} (x) - sin^{3} (x) cos (x) = 0

Factor

sin^{3} (x) - sin^{3} (x) cos (x) : - sin^{3} (x) (cos (x) - 1)

sin^{3} (x) - sin^{3} (x) cos (x)

Factor out common term

- sin^{3} (x)

= - sin^{3} (x) (- 1 + cos (x))

- sin^{3} (x) (cos (x) - 1) = 0

Solving each part separately

sin^{3} (x) = 0 or cos (x) - 1 = 0

sin^{3} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{3} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Move

1

to the right side

cos (x) - 1 = 0

Add

1

to both sides

cos (x) - 1 + 1 = 0 + 1

Simplify

cos (x) = 1

cos (x) = 1

General solutions for

cos (x) = 1

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

x = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Graph

Popular Examples

sec^2(x)-1=tan(x)

sec^{2} (x) - 1 = tan (x)

tan(x/4)=1

tan (\frac{x}{4}) = 1

2sin(2x)=cos(x)

2 sin (2 x) = cos (x)

(tan(θ)+1)(sec(θ)-2)=0

(tan (θ) + 1) (sec (θ) - 2) = 0

sin(3x)=2sin(x)

sin (3 x) = 2 sin (x)

Frequently Asked Questions (FAQ)

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