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Popular >

sin^3(x)-0.25sin(x)=0

Solution

sin^{3} (x) - 0.25 sin (x) = 0

Solution

x = 2 πn, x = π + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{3} (x) - 0.25 sin (x) = 0

Solve by substitution

sin^{3} (x) - 0.25 sin (x) = 0

Let:

sin (x) = u

u^{3} - 0.25 u = 0

u^{3} - 0.25 u = 0 : u = 0, u = - 0.5, u = 0.5

u^{3} - 0.25 u = 0

Factor

u^{3} - 0.25 u : u (u + 0.25) (u - 0.25)

u^{3} - 0.25 u

Factor out common term

u : u (u^{2} - 0.25)

u^{3} - 0.25 u

Apply exponent rule:

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

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

= u^{2} u - 0.25 u

Factor out common term

u

= u (u^{2} - 0.25)

= u (u^{2} - 0.25)

Factor

u^{2} - 0.25 : (u + 0.25) (u - 0.25)

u^{2} - 0.25

Apply radical rule:

a = (a)^{2}

0.25 = (0.25)^{2}

= u^{2} - (0.25)^{2}

Apply Difference of Two Squares Formula:

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

u^{2} - (0.25)^{2} = (u + 0.25) (u - 0.25)

= (u + 0.25) (u - 0.25)

= u (u + 0.25) (u - 0.25)

u (u + 0.25) (u - 0.25) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or u + 0.25 = 0 or u - 0.25 = 0

Solve

u + 0.25 = 0 : u = - 0.5

u + 0.25 = 0

Move

0.25

to the right side

u + 0.25 = 0

Subtract

0.25

from both sides

u + 0.25 - 0.25 = 0 - 0.25

Simplify

u = - 0.5

u = - 0.5

Solve

u - 0.25 = 0 : u = 0.5

u - 0.25 = 0

Move

0.25

to the right side

u - 0.25 = 0

Add

0.25

to both sides

u - 0.25 + 0.25 = 0 + 0.25

Simplify

u = 0.5

u = 0.5

The solutions are

u = 0, u = - 0.5, u = 0.5

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - 0.5, sin (x) = 0.5

sin (x) = 0, sin (x) = - 0.5, sin (x) = 0.5

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) = - 0.5 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = - 0.5

General solutions for

sin (x) = - 0.5

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

sin (x) = 0.5 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = 0.5

General solutions for

sin (x) = 0.5

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 = 2 πn, x = π + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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