What is the general solution for sin(x)+sin^2(x)+sin^3(x)=1 ?

The general solution for sin(x)+sin^2(x)+sin^3(x)=1 is x=0.57482…+2pin,x=pi-0.57482…+2pin

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Popular Trigonometry >

sin(x)+sin^2(x)+sin^3(x)=1

Solution

sin (x) + sin^{2} (x) + sin^{3} (x) = 1

Solution

x = 0.57482 \dots + 2 πn, x = π - 0.57482 \dots + 2 πn

Degrees

x = 32.93512 \dots^{\circ} + 36 0^{\circ} n, x = 147.06487 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) + sin^{2} (x) + sin^{3} (x) = 1

Solve by substitution

sin (x) + sin^{2} (x) + sin^{3} (x) = 1

Let:

sin (x) = u

u + u^{2} + u^{3} = 1

u + u^{2} + u^{3} = 1 : u \approx 0.54368 \dots

u + u^{2} + u^{3} = 1

Move

1

to the left side

u + u^{2} + u^{3} = 1

Subtract

1

from both sides

u + u^{2} + u^{3} - 1 = 1 - 1

Simplify

u + u^{2} + u^{3} - 1 = 0

u + u^{2} + u^{3} - 1 = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{3} + u^{2} + u - 1 = 0

Find one solution for

u^{3} + u^{2} + u - 1 = 0

using Newton-Raphson:

u \approx 0.54368 \dots

u^{3} + u^{2} + u - 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} + u^{2} + u - 1

Find

f^{^{'}} (u) : 3 u^{2} + 2 u + 1

\frac{d}{d u} (u^{3} + u^{2} + u - 1)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{3}) + \frac{d}{d u} (u^{2}) + \frac{d u}{d u} - \frac{d}{d u} (1)

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

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

Apply the Power Rule:

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

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

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

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

Apply the Power Rule:

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

= 2 u^{2 - 1}

Simplify

= 2 u

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

\frac{d u}{d u}

Apply the common derivative:

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

= 1

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

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

Derivative of a constant:

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

= 0

= 3 u^{2} + 2 u + 1 - 0

Simplify

= 3 u^{2} + 2 u + 1

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.66666 \dots : Δ u_{1} = 0.33333 \dots

f (u_{0}) = 1^{3} + 1^{2} + 1 - 1 = 2

f^{^{'}} (u_{0}) = 3 \cdot 1^{2} + 2 \cdot 1 + 1 = 6

u_{1} = 0.66666 \dots

Δ u_{1} = ∣ 0.66666 \dots - 1 ∣ = 0.33333 \dots

Δ u_{1} = 0.33333 \dots

u_{2} = 0.55555 \dots : Δ u_{2} = 0.11111 \dots

f (u_{1}) = 0.66666 \dots^{3} + 0.66666 \dots^{2} + 0.66666 \dots - 1 = 0.40740 \dots

f^{^{'}} (u_{1}) = 3 \cdot 0.66666 \dots^{2} + 2 \cdot 0.66666 \dots + 1 = 3.66666 \dots

u_{2} = 0.55555 \dots

Δ u_{2} = ∣ 0.55555 \dots - 0.66666 \dots ∣ = 0.11111 \dots

Δ u_{2} = 0.11111 \dots

u_{3} = 0.54381 \dots : Δ u_{3} = 0.01174 \dots

f (u_{2}) = 0.55555 \dots^{3} + 0.55555 \dots^{2} + 0.55555 \dots - 1 = 0.03566 \dots

f^{^{'}} (u_{2}) = 3 \cdot 0.55555 \dots^{2} + 2 \cdot 0.55555 \dots + 1 = 3.03703 \dots

u_{3} = 0.54381 \dots

Δ u_{3} = ∣ 0.54381 \dots - 0.55555 \dots ∣ = 0.01174 \dots

Δ u_{3} = 0.01174 \dots

u_{4} = 0.54368 \dots : Δ u_{4} = 0.00012 \dots

f (u_{3}) = 0.54381 \dots^{3} + 0.54381 \dots^{2} + 0.54381 \dots - 1 = 0.00036 \dots

f^{^{'}} (u_{3}) = 3 \cdot 0.54381 \dots^{2} + 2 \cdot 0.54381 \dots + 1 = 2.97481 \dots

u_{4} = 0.54368 \dots

Δ u_{4} = ∣ 0.54368 \dots - 0.54381 \dots ∣ = 0.00012 \dots

Δ u_{4} = 0.00012 \dots

u_{5} = 0.54368 \dots : Δ u_{5} = 1.34021 E - 8

f (u_{4}) = 0.54368 \dots^{3} + 0.54368 \dots^{2} + 0.54368 \dots - 1 = 3.98601 E - 8

f^{^{'}} (u_{4}) = 3 \cdot 0.54368 \dots^{2} + 2 \cdot 0.54368 \dots + 1 = 2.97417 \dots

u_{5} = 0.54368 \dots

Δ u_{5} = ∣ 0.54368 \dots - 0.54368 \dots ∣ = 1.34021 E - 8

Δ u_{5} = 1.34021 E - 8

u \approx 0.54368 \dots

Apply long division:

\frac{u ^{3} + u ^{2} + u - 1}{u - 0.54368 \dots} = u^{2} + 1.54368 \dots u + 1.83928 \dots

u^{2} + 1.54368 \dots u + 1.83928 \dots \approx 0

Find one solution for

u^{2} + 1.54368 \dots u + 1.83928 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

u^{2} + 1.54368 \dots u + 1.83928 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} + 1.54368 \dots u + 1.83928 \dots

Find

f^{^{'}} (u) : 2 u + 1.54368 \dots

\frac{d}{d u} (u^{2} + 1.54368 \dots u + 1.83928 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (u^{2}) + \frac{d}{d u} (1.54368 \dots u) + \frac{d}{d u} (1.83928 \dots)

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

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

Apply the Power Rule:

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

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (1.54368 \dots u) = 1.54368 \dots

\frac{d}{d u} (1.54368 \dots u)

Take the constant out:

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

= 1.54368 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.54368 \dots \cdot 1

Simplify

= 1.54368 \dots

\frac{d}{d u} (1.83928 \dots) = 0

\frac{d}{d u} (1.83928 \dots)

Derivative of a constant:

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

= 0

= 2 u + 1.54368 \dots + 0

Simplify

= 2 u + 1.54368 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.83928 \dots : Δ u_{1} = 2.83928 \dots

f (u_{0}) = (- 1)^{2} + 1.54368 \dots (- 1) + 1.83928 \dots = 1.29559 \dots

f^{^{'}} (u_{0}) = 2 (- 1) + 1.54368 \dots = - 0.45631 \dots

u_{1} = 1.83928 \dots

Δ u_{1} = ∣ 1.83928 \dots - (- 1) ∣ = 2.83928 \dots

Δ u_{1} = 2.83928 \dots

u_{2} = 0.29559 \dots : Δ u_{2} = 1.54368 \dots

f (u_{1}) = 1.83928 \dots^{2} + 1.54368 \dots \cdot 1.83928 \dots + 1.83928 \dots = 8.06154 \dots

f^{^{'}} (u_{1}) = 2 \cdot 1.83928 \dots + 1.54368 \dots = 5.22226 \dots

u_{2} = 0.29559 \dots

Δ u_{2} = ∣ 0.29559 \dots - 1.83928 \dots ∣ = 1.54368 \dots

Δ u_{2} = 1.54368 \dots

u_{3} = - 0.82061 \dots : Δ u_{3} = 1.11620 \dots

f (u_{2}) = 0.29559 \dots^{2} + 1.54368 \dots \cdot 0.29559 \dots + 1.83928 \dots = 2.38297 \dots

f^{^{'}} (u_{2}) = 2 \cdot 0.29559 \dots + 1.54368 \dots = 2.13488 \dots

u_{3} = - 0.82061 \dots

Δ u_{3} = ∣ - 0.82061 \dots - 0.29559 \dots ∣ = 1.11620 \dots

Δ u_{3} = 1.11620 \dots

u_{4} = 11.95386 \dots : Δ u_{4} = 12.77447 \dots

f (u_{3}) = (- 0.82061 \dots)^{2} + 1.54368 \dots (- 0.82061 \dots) + 1.83928 \dots = 1.24592 \dots

f^{^{'}} (u_{3}) = 2 (- 0.82061 \dots) + 1.54368 \dots = - 0.09753 \dots

u_{4} = 11.95386 \dots

Δ u_{4} = ∣ 11.95386 \dots - (- 0.82061 \dots) ∣ = 12.77447 \dots

Δ u_{4} = 12.77447 \dots

u_{5} = 5.54214 \dots : Δ u_{5} = 6.41171 \dots

f (u_{4}) = 11.95386 \dots^{2} + 1.54368 \dots \cdot 11.95386 \dots + 1.83928 \dots = 163.18717 \dots

f^{^{'}} (u_{4}) = 2 \cdot 11.95386 \dots + 1.54368 \dots = 25.45141 \dots

u_{5} = 5.54214 \dots

Δ u_{5} = ∣ 5.54214 \dots - 11.95386 \dots ∣ = 6.41171 \dots

Δ u_{5} = 6.41171 \dots

u_{6} = 2.28667 \dots : Δ u_{6} = 3.25547 \dots

f (u_{5}) = 5.54214 \dots^{2} + 1.54368 \dots \cdot 5.54214 \dots + 1.83928 \dots = 41.11006 \dots

f^{^{'}} (u_{5}) = 2 \cdot 5.54214 \dots + 1.54368 \dots = 12.62798 \dots

u_{6} = 2.28667 \dots

Δ u_{6} = ∣ 2.28667 \dots - 5.54214 \dots ∣ = 3.25547 \dots

Δ u_{6} = 3.25547 \dots

u_{7} = 0.55412 \dots : Δ u_{7} = 1.73255 \dots

f (u_{6}) = 2.28667 \dots^{2} + 1.54368 \dots \cdot 2.28667 \dots + 1.83928 \dots = 10.59809 \dots

f^{^{'}} (u_{6}) = 2 \cdot 2.28667 \dots + 1.54368 \dots = 6.11704 \dots

u_{7} = 0.55412 \dots

Δ u_{7} = ∣ 0.55412 \dots - 2.28667 \dots ∣ = 1.73255 \dots

Δ u_{7} = 1.73255 \dots

u_{8} = - 0.57777 \dots : Δ u_{8} = 1.13190 \dots

f (u_{7}) = 0.55412 \dots^{2} + 1.54368 \dots \cdot 0.55412 \dots + 1.83928 \dots = 3.00173 \dots

f^{^{'}} (u_{7}) = 2 \cdot 0.55412 \dots + 1.54368 \dots = 2.65193 \dots

u_{8} = - 0.57777 \dots

Δ u_{8} = ∣ - 0.57777 \dots - 0.55412 \dots ∣ = 1.13190 \dots

Δ u_{8} = 1.13190 \dots

u_{9} = - 3.87872 \dots : Δ u_{9} = 3.30094 \dots

f (u_{8}) = (- 0.57777 \dots)^{2} + 1.54368 \dots (- 0.57777 \dots) + 1.83928 \dots = 1.28120 \dots

f^{^{'}} (u_{8}) = 2 (- 0.57777 \dots) + 1.54368 \dots = 0.38813 \dots

u_{9} = - 3.87872 \dots

Δ u_{9} = ∣ - 3.87872 \dots - (- 0.57777 \dots) ∣ = 3.30094 \dots

Δ u_{9} = 3.30094 \dots

u_{10} = - 2.12515 \dots : Δ u_{10} = 1.75356 \dots

f (u_{9}) = (- 3.87872 \dots)^{2} + 1.54368 \dots (- 3.87872 \dots) + 1.83928 \dots = 10.89622 \dots

f^{^{'}} (u_{9}) = 2 (- 3.87872 \dots) + 1.54368 \dots = - 6.21375 \dots

u_{10} = - 2.12515 \dots

Δ u_{10} = ∣ - 2.12515 \dots - (- 3.87872 \dots) ∣ = 1.75356 \dots

Δ u_{10} = 1.75356 \dots

u_{11} = - 0.98905 \dots : Δ u_{11} = 1.13610 \dots

f (u_{10}) = (- 2.12515 \dots)^{2} + 1.54368 \dots (- 2.12515 \dots) + 1.83928 \dots = 3.07499 \dots

f^{^{'}} (u_{10}) = 2 (- 2.12515 \dots) + 1.54368 \dots = - 2.70662 \dots

u_{11} = - 0.98905 \dots

Δ u_{11} = ∣ - 0.98905 \dots - (- 2.12515 \dots) ∣ = 1.13610 \dots

Δ u_{11} = 1.13610 \dots

u_{12} = 1.98207 \dots : Δ u_{12} = 2.97113 \dots

f (u_{11}) = (- 0.98905 \dots)^{2} + 1.54368 \dots (- 0.98905 \dots) + 1.83928 \dots = 1.29072 \dots

f^{^{'}} (u_{11}) = 2 (- 0.98905 \dots) + 1.54368 \dots = - 0.43442 \dots

u_{12} = 1.98207 \dots

Δ u_{12} = ∣ 1.98207 \dots - (- 0.98905 \dots) ∣ = 2.97113 \dots

Δ u_{12} = 2.97113 \dots

u_{13} = 0.37934 \dots : Δ u_{13} = 1.60273 \dots

f (u_{12}) = 1.98207 \dots^{2} + 1.54368 \dots \cdot 1.98207 \dots + 1.83928 \dots = 8.82763 \dots

f^{^{'}} (u_{12}) = 2 \cdot 1.98207 \dots + 1.54368 \dots = 5.50784 \dots

u_{13} = 0.37934 \dots

Δ u_{13} = ∣ 0.37934 \dots - 1.98207 \dots ∣ = 1.60273 \dots

Δ u_{13} = 1.60273 \dots

u_{14} = - 0.73636 \dots : Δ u_{14} = 1.11570 \dots

f (u_{13}) = 0.37934 \dots^{2} + 1.54368 \dots \cdot 0.37934 \dots + 1.83928 \dots = 2.56876 \dots

f^{^{'}} (u_{13}) = 2 \cdot 0.37934 \dots + 1.54368 \dots = 2.30236 \dots

u_{14} = - 0.73636 \dots

Δ u_{14} = ∣ - 0.73636 \dots - 0.37934 \dots ∣ = 1.11570 \dots

Δ u_{14} = 1.11570 \dots

u_{15} = - 18.27956 \dots : Δ u_{15} = 17.54320 \dots

f (u_{14}) = (- 0.73636 \dots)^{2} + 1.54368 \dots (- 0.73636 \dots) + 1.83928 \dots = 1.24480 \dots

f^{^{'}} (u_{14}) = 2 (- 0.73636 \dots) + 1.54368 \dots = 0.07095 \dots

u_{15} = - 18.27956 \dots

Δ u_{15} = ∣ - 18.27956 \dots - (- 0.73636 \dots) ∣ = 17.54320 \dots

Δ u_{15} = 17.54320 \dots

Cannot find solution

The solution is

u \approx 0.54368 \dots

Substitute back

u = sin (x)

sin (x) \approx 0.54368 \dots

sin (x) \approx 0.54368 \dots

sin (x) = 0.54368 \dots : x = arcsin (0.54368 \dots) + 2 πn, x = π - arcsin (0.54368 \dots) + 2 πn

sin (x) = 0.54368 \dots

Apply trig inverse properties

sin (x) = 0.54368 \dots

General solutions for

sin (x) = 0.54368 \dots

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

x = arcsin (0.54368 \dots) + 2 πn, x = π - arcsin (0.54368 \dots) + 2 πn

x = arcsin (0.54368 \dots) + 2 πn, x = π - arcsin (0.54368 \dots) + 2 πn

Combine all the solutions

x = arcsin (0.54368 \dots) + 2 πn, x = π - arcsin (0.54368 \dots) + 2 πn

Show solutions in decimal form

x = 0.57482 \dots + 2 πn, x = π - 0.57482 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)+sin^2(x)+sin^3(x)=1 ?
The general solution for sin(x)+sin^2(x)+sin^3(x)=1 is x=0.57482…+2pin,x=pi-0.57482…+2pin