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Popular Trigonometry >

sin^3(a)=3sin(a)-4sin^3(a)

Solution

sin^{3} (a) = 3 sin (a) - 4 sin^{3} (a)

Solution

a = 2 πn, a = π + 2 πn, a = - 0.88607 \dots + 2 πn, a = π + 0.88607 \dots + 2 πn, a = 0.88607 \dots + 2 πn, a = π - 0.88607 \dots + 2 πn

Degrees

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = - 50.76847 \dots^{\circ} + 36 0^{\circ} n, a = 230.76847 \dots^{\circ} + 36 0^{\circ} n, a = 50.76847 \dots^{\circ} + 36 0^{\circ} n, a = 129.23152 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{3} (a) = 3 sin (a) - 4 sin^{3} (a)

Solve by substitution

sin^{3} (a) = 3 sin (a) - 4 sin^{3} (a)

Let:

sin (a) = u

u^{3} = 3 u - 4 u^{3}

u^{3} = 3 u - 4 u^{3} : u = 0, u = - \frac{3}{5}, u = \frac{3}{5}

u^{3} = 3 u - 4 u^{3}

Switch sides

3 u - 4 u^{3} = u^{3}

Move

u^{3}

to the left side

3 u - 4 u^{3} = u^{3}

Subtract

u^{3}

from both sides

3 u - 4 u^{3} - u^{3} = u^{3} - u^{3}

Simplify

3 u - 5 u^{3} = 0

3 u - 5 u^{3} = 0

Factor

3 u - 5 u^{3} : - u (5 u + 3) (5 u - 3)

3 u - 5 u^{3}

Factor out common term

- u : - u (5 u^{2} - 3)

- 5 u^{3} + 3 u

Apply exponent rule:

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

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

= - 5 u^{2} u + 3 u

Factor out common term

- u

= - u (5 u^{2} - 3)

= - u (5 u^{2} - 3)

Factor

5 u^{2} - 3 : (5 u + 3) (5 u - 3)

5 u^{2} - 3

Rewrite

5 u^{2} - 3

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

5 u^{2} - 3

Apply radical rule:

a = (a)^{2}

5 = (5)^{2}

= (5)^{2} u^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (5)^{2} u^{2} - (3)^{2}

Apply exponent rule:

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

(5)^{2} u^{2} = (5 u)^{2}

= (5 u)^{2} - (3)^{2}

= (5 u)^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

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

(5 u)^{2} - (3)^{2} = (5 u + 3) (5 u - 3)

= (5 u + 3) (5 u - 3)

= - u (5 u + 3) (5 u - 3)

- u (5 u + 3) (5 u - 3) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

u = 0 or 5 u + 3 = 0 or 5 u - 3 = 0

Solve

5 u + 3 = 0 : u = - \frac{3}{5}

5 u + 3 = 0

Move

3

to the right side

5 u + 3 = 0

Subtract

3

from both sides

5 u + 3 - 3 = 0 - 3

Simplify

5 u = - 3

5 u = - 3

Divide both sides by

5

5 u = - 3

Divide both sides by

5

\frac{5 u}{5} = \frac{- 3}{5}

Simplify

\frac{5 u}{5} = \frac{- 3}{5}

Simplify

\frac{5 u}{5} : u

\frac{5 u}{5}

Cancel the common factor:

5

= u

Simplify

\frac{- 3}{5} : - \frac{3}{5}

\frac{- 3}{5}

Apply the fraction rule:

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

= - \frac{3}{5}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{5}

u = - \frac{3}{5}

u = - \frac{3}{5}

u = - \frac{3}{5}

Solve

5 u - 3 = 0 : u = \frac{3}{5}

5 u - 3 = 0

Move

3

to the right side

5 u - 3 = 0

Add

3

to both sides

5 u - 3 + 3 = 0 + 3

Simplify

5 u = 3

5 u = 3

Divide both sides by

5

5 u = 3

Divide both sides by

5

\frac{5 u}{5} = \frac{3}{5}

Simplify

\frac{5 u}{5} = \frac{3}{5}

Simplify

\frac{5 u}{5} : u

\frac{5 u}{5}

Cancel the common factor:

5

= u

Simplify

\frac{3}{5} : \frac{3}{5}

\frac{3}{5}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{5}

u = \frac{3}{5}

u = \frac{3}{5}

u = \frac{3}{5}

The solutions are

u = 0, u = - \frac{3}{5}, u = \frac{3}{5}

Substitute back

u = sin (a)

sin (a) = 0, sin (a) = - \frac{3}{5}, sin (a) = \frac{3}{5}

sin (a) = 0, sin (a) = - \frac{3}{5}, sin (a) = \frac{3}{5}

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

General solutions for

sin (a) = 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}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Solve

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

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

sin (a) = - \frac{3}{5}

Apply trig inverse properties

sin (a) = - \frac{3}{5}

General solutions for

sin (a) = - \frac{3}{5}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

a = arcsin (- \frac{3}{5}) + 2 πn, a = π + arcsin (\frac{3}{5}) + 2 πn

a = arcsin (- \frac{3}{5}) + 2 πn, a = π + arcsin (\frac{3}{5}) + 2 πn

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

sin (a) = \frac{3}{5}

Apply trig inverse properties

sin (a) = \frac{3}{5}

General solutions for

sin (a) = \frac{3}{5}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

a = arcsin (\frac{3}{5}) + 2 πn, a = π - arcsin (\frac{3}{5}) + 2 πn

a = arcsin (\frac{3}{5}) + 2 πn, a = π - arcsin (\frac{3}{5}) + 2 πn

Combine all the solutions

a = 2 πn, a = π + 2 πn, a = arcsin (- \frac{3}{5}) + 2 πn, a = π + arcsin (\frac{3}{5}) + 2 πn, a = arcsin (\frac{3}{5}) + 2 πn, a = π - arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

a = 2 πn, a = π + 2 πn, a = - 0.88607 \dots + 2 πn, a = π + 0.88607 \dots + 2 πn, a = 0.88607 \dots + 2 πn, a = π - 0.88607 \dots + 2 πn

Graph

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