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Popular >

12sin^3(x)-9sin(x)=0

Solution

12 sin^{3} (x) - 9 sin (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Solution steps

12 sin^{3} (x) - 9 sin (x) = 0

Solve by substitution

12 sin^{3} (x) - 9 sin (x) = 0

Let:

sin (x) = u

12 u^{3} - 9 u = 0

12 u^{3} - 9 u = 0 : u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

12 u^{3} - 9 u = 0

Factor

12 u^{3} - 9 u : 3 u (2 u + 3) (2 u - 3)

12 u^{3} - 9 u

Factor out common term

3 u : 3 u (4 u^{2} - 3)

12 u^{3} - 9 u

Apply exponent rule:

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

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

= 12 u^{2} u - 9 u

Rewrite

9

3 \cdot 3

Rewrite

12

3 \cdot 4

= 3 \cdot 4 u^{2} u - 3 \cdot 3 u

Factor out common term

3 u

= 3 u (4 u^{2} - 3)

= 3 u (4 u^{2} - 3)

Factor

4 u^{2} - 3 : (2 u + 3) (2 u - 3)

4 u^{2} - 3

Rewrite

4 u^{2} - 3

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

4 u^{2} - 3

Rewrite

4

2^{2}

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

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

2^{2} u^{2} = (2 u)^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

= (2 u + 3) (2 u - 3)

= 3 u (2 u + 3) (2 u - 3)

3 u (2 u + 3) (2 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 2 u + 3 = 0 or 2 u - 3 = 0

Solve

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

Move

3

to the right side

2 u + 3 = 0

Subtract

3

from both sides

2 u + 3 - 3 = 0 - 3

Simplify

2 u = - 3

2 u = - 3

Divide both sides by

2

2 u = - 3

Divide both sides by

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplify

u = - \frac{3}{2}

u = - \frac{3}{2}

Solve

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

Move

3

to the right side

2 u - 3 = 0

Add

3

to both sides

2 u - 3 + 3 = 0 + 3

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

\frac{2 u}{2} = \frac{3}{2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

The solutions are

u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

Substitute back

u = sin (x)

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

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

sin (x) = - \frac{3}{2} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sin (x) = - \frac{3}{2}

General solutions for

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

sin (x) = \frac{3}{2}

General solutions for

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

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

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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