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Popular >

8sin^3(x)-6sin(x)+1=0

Solution

8 sin^{3} (x) - 6 sin (x) + 1 = 0

Solution

x = 0.17453 \dots + 2 πn, x = π - 0.17453 \dots + 2 πn, x = 0.87266 \dots + 2 πn, x = π - 0.87266 \dots + 2 πn, x = - 1.22173 \dots + 2 πn, x = π + 1.22173 \dots + 2 πn

Degrees

x = 1 0^{\circ} + 36 0^{\circ} n, x = 17 0^{\circ} + 36 0^{\circ} n, x = 5 0^{\circ} + 36 0^{\circ} n, x = 13 0^{\circ} + 36 0^{\circ} n, x = - 7 0^{\circ} + 36 0^{\circ} n, x = 25 0^{\circ} + 36 0^{\circ} n

Solution steps

8 sin^{3} (x) - 6 sin (x) + 1 = 0

Solve by substitution

8 sin^{3} (x) - 6 sin (x) + 1 = 0

Let:

sin (x) = u

8 u^{3} - 6 u + 1 = 0

8 u^{3} - 6 u + 1 = 0 : u \approx 0.17364 \dots, u \approx 0.76604 \dots, u \approx - 0.93969 \dots

8 u^{3} - 6 u + 1 = 0

Find one solution for

8 u^{3} - 6 u + 1 = 0

using Newton-Raphson:

u \approx 0.17364 \dots

8 u^{3} - 6 u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = 8 u^{3} - 6 u + 1

Find

f^{^{'}} (u) : 24 u^{2} - 6

\frac{d}{d u} (8 u^{3} - 6 u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (8 u^{3}) - \frac{d}{d u} (6 u) + \frac{d}{d u} (1)

\frac{d}{d u} (8 u^{3}) = 24 u^{2}

\frac{d}{d u} (8 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 8 \cdot 3 u^{3 - 1}

Simplify

= 24 u^{2}

\frac{d}{d u} (6 u) = 6

\frac{d}{d u} (6 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 6 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 6 \cdot 1

Simplify

= 6

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 24 u^{2} - 6 + 0

Simplify

= 24 u^{2} - 6

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.16666 \dots : Δ u_{1} = 0.16666 \dots

f (u_{0}) = 8 \cdot 0^{3} - 6 \cdot 0 + 1 = 1

f^{^{'}} (u_{0}) = 24 \cdot 0^{2} - 6 = - 6

u_{1} = 0.16666 \dots

Δ u_{1} = ∣ 0.16666 \dots - 0 ∣ = 0.16666 \dots

Δ u_{1} = 0.16666 \dots

u_{2} = 0.17361 \dots : Δ u_{2} = 0.00694 \dots

f (u_{1}) = 8 \cdot 0.16666 \dots^{3} - 6 \cdot 0.16666 \dots + 1 = 0.03703 \dots

f^{^{'}} (u_{1}) = 24 \cdot 0.16666 \dots^{2} - 6 = - 5.33333 \dots

u_{2} = 0.17361 \dots

Δ u_{2} = ∣ 0.17361 \dots - 0.16666 \dots ∣ = 0.00694 \dots

Δ u_{2} = 0.00694 \dots

u_{3} = 0.17364 \dots : Δ u_{3} = 0.00003 \dots

f (u_{2}) = 8 \cdot 0.17361 \dots^{3} - 6 \cdot 0.17361 \dots + 1 = 0.00019 \dots

f^{^{'}} (u_{2}) = 24 \cdot 0.17361 \dots^{2} - 6 = - 5.27662 \dots

u_{3} = 0.17364 \dots

Δ u_{3} = ∣ 0.17364 \dots - 0.17361 \dots ∣ = 0.00003 \dots

Δ u_{3} = 0.00003 \dots

u_{4} = 0.17364 \dots : Δ u_{4} = 1.085 E - 9

f (u_{3}) = 8 \cdot 0.17364 \dots^{3} - 6 \cdot 0.17364 \dots + 1 = 5.72478 E - 9

f^{^{'}} (u_{3}) = 24 \cdot 0.17364 \dots^{2} - 6 = - 5.27631 \dots

u_{4} = 0.17364 \dots

Δ u_{4} = ∣ 0.17364 \dots - 0.17364 \dots ∣ = 1.085 E - 9

Δ u_{4} = 1.085 E - 9

u \approx 0.17364 \dots

Apply long division:

\frac{8 u ^{3} - 6 u + 1}{u - 0.17364 \dots} = 8 u^{2} + 1.38918 \dots u - 5.75877 \dots

8 u^{2} + 1.38918 \dots u - 5.75877 \dots \approx 0

Find one solution for

8 u^{2} + 1.38918 \dots u - 5.75877 \dots = 0

using Newton-Raphson:

u \approx 0.76604 \dots

8 u^{2} + 1.38918 \dots u - 5.75877 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 8 u^{2} + 1.38918 \dots u - 5.75877 \dots

Find

f^{^{'}} (u) : 16 u + 1.38918 \dots

\frac{d}{d u} (8 u^{2} + 1.38918 \dots u - 5.75877 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (8 u^{2}) + \frac{d}{d u} (1.38918 \dots u) - \frac{d}{d u} (5.75877 \dots)

\frac{d}{d u} (8 u^{2}) = 16 u

\frac{d}{d u} (8 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 8 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 8 \cdot 2 u^{2 - 1}

Simplify

= 16 u

\frac{d}{d u} (1.38918 \dots u) = 1.38918 \dots

\frac{d}{d u} (1.38918 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 1.38918 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 1.38918 \dots \cdot 1

Simplify

= 1.38918 \dots

\frac{d}{d u} (5.75877 \dots) = 0

\frac{d}{d u} (5.75877 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 16 u + 1.38918 \dots - 0

Simplify

= 16 u + 1.38918 \dots

Let

u_{0} = 4

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 2.04557 \dots : Δ u_{1} = 1.95442 \dots

f (u_{0}) = 8 \cdot 4^{2} + 1.38918 \dots \cdot 4 - 5.75877 \dots = 127.79797 \dots

f^{^{'}} (u_{0}) = 16 \cdot 4 + 1.38918 \dots = 65.38918 \dots

u_{1} = 2.04557 \dots

Δ u_{1} = ∣ 2.04557 \dots - 4 ∣ = 1.95442 \dots

Δ u_{1} = 1.95442 \dots

u_{2} = 1.14993 \dots : Δ u_{2} = 0.89564 \dots

f (u_{1}) = 8 \cdot 2.04557 \dots^{2} + 1.38918 \dots \cdot 2.04557 \dots - 5.75877 \dots = 30.55807 \dots

f^{^{'}} (u_{1}) = 16 \cdot 2.04557 \dots + 1.38918 \dots = 34.11845 \dots

u_{2} = 1.14993 \dots

Δ u_{2} = ∣ 1.14993 \dots - 2.04557 \dots ∣ = 0.89564 \dots

Δ u_{2} = 0.89564 \dots

u_{3} = 0.82562 \dots : Δ u_{3} = 0.32430 \dots

f (u_{2}) = 8 \cdot 1.14993 \dots^{2} + 1.38918 \dots \cdot 1.14993 \dots - 5.75877 \dots = 6.41746 \dots

f^{^{'}} (u_{2}) = 16 \cdot 1.14993 \dots + 1.38918 \dots = 19.78811 \dots

u_{3} = 0.82562 \dots

Δ u_{3} = ∣ 0.82562 \dots - 1.14993 \dots ∣ = 0.32430 \dots

Δ u_{3} = 0.32430 \dots

u_{4} = 0.76798 \dots : Δ u_{4} = 0.05763 \dots

f (u_{3}) = 8 \cdot 0.82562 \dots^{2} + 1.38918 \dots \cdot 0.82562 \dots - 5.75877 \dots = 0.84141 \dots

f^{^{'}} (u_{3}) = 16 \cdot 0.82562 \dots + 1.38918 \dots = 14.59916 \dots

u_{4} = 0.76798 \dots

Δ u_{4} = ∣ 0.76798 \dots - 0.82562 \dots ∣ = 0.05763 \dots

Δ u_{4} = 0.05763 \dots

u_{5} = 0.76604 \dots : Δ u_{5} = 0.00194 \dots

f (u_{4}) = 8 \cdot 0.76798 \dots^{2} + 1.38918 \dots \cdot 0.76798 \dots - 5.75877 \dots = 0.02657 \dots

f^{^{'}} (u_{4}) = 16 \cdot 0.76798 \dots + 1.38918 \dots = 13.67701 \dots

u_{5} = 0.76604 \dots

Δ u_{5} = ∣ 0.76604 \dots - 0.76798 \dots ∣ = 0.00194 \dots

Δ u_{5} = 0.00194 \dots

u_{6} = 0.76604 \dots : Δ u_{6} = 2.21312 E - 6

f (u_{5}) = 8 \cdot 0.76604 \dots^{2} + 1.38918 \dots \cdot 0.76604 \dots - 5.75877 \dots = 0.00003 \dots

f^{^{'}} (u_{5}) = 16 \cdot 0.76604 \dots + 1.38918 \dots = 13.64593 \dots

u_{6} = 0.76604 \dots

Δ u_{6} = ∣ 0.76604 \dots - 0.76604 \dots ∣ = 2.21312 E - 6

Δ u_{6} = 2.21312 E - 6

u_{7} = 0.76604 \dots : Δ u_{7} = 2.87147 E - 12

f (u_{6}) = 8 \cdot 0.76604 \dots^{2} + 1.38918 \dots \cdot 0.76604 \dots - 5.75877 \dots = 3.91838 E - 11

f^{^{'}} (u_{6}) = 16 \cdot 0.76604 \dots + 1.38918 \dots = 13.64589 \dots

u_{7} = 0.76604 \dots

Δ u_{7} = ∣ 0.76604 \dots - 0.76604 \dots ∣ = 2.87147 E - 12

Δ u_{7} = 2.87147 E - 12

u \approx 0.76604 \dots

Apply long division:

\frac{8 u ^{2} + 1.38918 \dots u - 5.75877 \dots}{u - 0.76604 \dots} = 8 u + 7.51754 \dots

8 u + 7.51754 \dots \approx 0

u \approx - 0.93969 \dots

The solutions are

u \approx 0.17364 \dots, u \approx 0.76604 \dots, u \approx - 0.93969 \dots

Substitute back

u = sin (x)

sin (x) \approx 0.17364 \dots, sin (x) \approx 0.76604 \dots, sin (x) \approx - 0.93969 \dots

sin (x) \approx 0.17364 \dots, sin (x) \approx 0.76604 \dots, sin (x) \approx - 0.93969 \dots

sin (x) = 0.17364 \dots : x = arcsin (0.17364 \dots) + 2 πn, x = π - arcsin (0.17364 \dots) + 2 πn

sin (x) = 0.17364 \dots

Apply trig inverse properties

sin (x) = 0.17364 \dots

General solutions for

sin (x) = 0.17364 \dots

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

x = arcsin (0.17364 \dots) + 2 πn, x = π - arcsin (0.17364 \dots) + 2 πn

x = arcsin (0.17364 \dots) + 2 πn, x = π - arcsin (0.17364 \dots) + 2 πn

sin (x) = 0.76604 \dots : x = arcsin (0.76604 \dots) + 2 πn, x = π - arcsin (0.76604 \dots) + 2 πn

sin (x) = 0.76604 \dots

Apply trig inverse properties

sin (x) = 0.76604 \dots

General solutions for

sin (x) = 0.76604 \dots

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

x = arcsin (0.76604 \dots) + 2 πn, x = π - arcsin (0.76604 \dots) + 2 πn

x = arcsin (0.76604 \dots) + 2 πn, x = π - arcsin (0.76604 \dots) + 2 πn

sin (x) = - 0.93969 \dots : x = arcsin (- 0.93969 \dots) + 2 πn, x = π + arcsin (0.93969 \dots) + 2 πn

sin (x) = - 0.93969 \dots

Apply trig inverse properties

sin (x) = - 0.93969 \dots

General solutions for

sin (x) = - 0.93969 \dots

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

x = arcsin (- 0.93969 \dots) + 2 πn, x = π + arcsin (0.93969 \dots) + 2 πn

x = arcsin (- 0.93969 \dots) + 2 πn, x = π + arcsin (0.93969 \dots) + 2 πn

Combine all the solutions

x = arcsin (0.17364 \dots) + 2 πn, x = π - arcsin (0.17364 \dots) + 2 πn, x = arcsin (0.76604 \dots) + 2 πn, x = π - arcsin (0.76604 \dots) + 2 πn, x = arcsin (- 0.93969 \dots) + 2 πn, x = π + arcsin (0.93969 \dots) + 2 πn

Show solutions in decimal form

x = 0.17453 \dots + 2 πn, x = π - 0.17453 \dots + 2 πn, x = 0.87266 \dots + 2 πn, x = π - 0.87266 \dots + 2 πn, x = - 1.22173 \dots + 2 πn, x = π + 1.22173 \dots + 2 πn

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