What is the general solution for sin(c)= 2/(pi(-cos(c)+1)) ?

The general solution for sin(c)= 2/(pi(-cos(c)+1)) is c=1.23822…+2pin,c=2.80812…+2pin

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sin(c)= 2/(pi(-cos(c)+1))

Solution

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

Solution

c = 1.23822 \dots + 2 πn, c = 2.80812 \dots + 2 πn

Degrees

c = 70.94503 \dots^{\circ} + 36 0^{\circ} n, c = 160.89345 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

Square both sides

sin^{2} (c) = (\frac{2}{π ( - cos ( c ) + 1 )})^{2}

Subtract

(\frac{2}{π ( - cos ( c ) + 1 )})^{2}

from both sides

sin^{2} (c) - \frac{4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}} = 0

Simplify

sin^{2} (c) - \frac{4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}} : \frac{π ^{2} sin ^{2} ( c ) ( - cos ( c ) + 1 ) ^{2} - 4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}}

sin^{2} (c) - \frac{4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}}

Convert element to fraction:

sin^{2} (c) = \frac{sin ^{2} ( c ) π ^{2} ( - cos ( c ) + 1 ) ^{2}}{π ^{2} ( - cos ( c ) + 1 ) ^{2}}

= \frac{sin ^{2} ( c ) π ^{2} ( - cos ( c ) + 1 ) ^{2}}{π ^{2} ( - cos ( c ) + 1 ) ^{2}} - \frac{4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ^{2} ( c ) π ^{2} ( - cos ( c ) + 1 ) ^{2} - 4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}}

\frac{π ^{2} sin ^{2} ( c ) ( - cos ( c ) + 1 ) ^{2} - 4}{π ^{2} ( - cos ( c ) + 1 ) ^{2}} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

π^{2} sin^{2} (c) (- cos (c) + 1)^{2} - 4 = 0

Rewrite using trig identities

- 4 + (1 - cos (c))^{2} sin^{2} (c) π^{2}

Use the Pythagorean identity:

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

sin^{2} (x) = 1 - cos^{2} (x)

= - 4 + (1 - cos (c))^{2} (1 - cos^{2} (c)) π^{2}

- 4 + (1 - cos (c))^{2} (1 - cos^{2} (c)) π^{2} = 0

Solve by substitution

- 4 + (1 - cos (c))^{2} (1 - cos^{2} (c)) π^{2} = 0

Let:

cos (c) = u

- 4 + (1 - u)^{2} (1 - u^{2}) π^{2} = 0

- 4 + (1 - u)^{2} (1 - u^{2}) π^{2} = 0 : u \approx 0.32647 \dots, u \approx - 0.94491 \dots

- 4 + (1 - u)^{2} (1 - u^{2}) π^{2} = 0

Expand

- 4 + (1 - u)^{2} (1 - u^{2}) π^{2} : - 4 + 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

- 4 + (1 - u)^{2} (1 - u^{2}) π^{2}

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

(1 - u)^{2}

Apply Perfect Square Formula:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 1, b = u

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

Simplify

1^{2} - 2 \cdot 1 \cdot u + u^{2} : 1 - 2 u + u^{2}

1^{2} - 2 \cdot 1 \cdot u + u^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= 1 - 2 u + u^{2}

= 1 - 2 u + u^{2}

= - 4 + π^{2} (u^{2} - 2 u + 1) (- u^{2} + 1)

= - 4 + π^{2} (1 - 2 u + u^{2}) (1 - u^{2})

Expand

(1 - 2 u + u^{2}) (1 - u^{2}) π^{2} : 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

Expand

(1 - 2 u + u^{2}) (1 - u^{2}) : 2 u^{3} - u^{4} - 2 u + 1

(1 - 2 u + u^{2}) (1 - u^{2})

Distribute parentheses

= 1 \cdot 1 + 1 \cdot (- u^{2}) + (- 2 u) \cdot 1 + (- 2 u) (- u^{2}) + u^{2} \cdot 1 + u^{2} (- u^{2})

Apply minus-plus rules

+ (- a) = - a, (- a) (- b) = ab

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

Simplify

1 \cdot 1 - 1 \cdot u^{2} - 2 \cdot 1 \cdot u + 2 u^{2} u + 1 \cdot u^{2} - u^{2} u^{2} : 2 u^{3} - u^{4} - 2 u + 1

1 \cdot 1 - 1 \cdot u^{2} - 2 \cdot 1 \cdot u + 2 u^{2} u + 1 \cdot u^{2} - u^{2} u^{2}

Group like terms

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

Add similar elements:

- 1 \cdot u^{2} + 1 \cdot u^{2} = 0

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

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

2 u^{2} u

Apply exponent rule:

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

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

= 2 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 2 u^{3}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

Apply exponent rule:

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

u^{2} u^{2} = u^{2 + 2}

= u^{2 + 2}

Add the numbers:

2 + 2 = 4

= u^{4}

2 \cdot 1 \cdot u = 2 u

2 \cdot 1 \cdot u

Multiply the numbers:

2 \cdot 1 = 2

= 2 u

1 \cdot 1 = 1

1 \cdot 1

Multiply the numbers:

1 \cdot 1 = 1

= 1

= 2 u^{3} - u^{4} - 2 u + 1

= 2 u^{3} - u^{4} - 2 u + 1

= π^{2} (2 u^{3} - u^{4} - 2 u + 1)

Expand

π^{2} (2 u^{3} - u^{4} - 2 u + 1) : 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

π^{2} (2 u^{3} - u^{4} - 2 u + 1)

Distribute parentheses

= π^{2} \cdot 2 u^{3} + π^{2} (- u^{4}) + π^{2} (- 2 u) + π^{2} \cdot 1

Apply minus-plus rules

+ (- a) = - a

= 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + 1 \cdot π^{2}

Multiply:

1 \cdot π^{2} = π^{2}

= 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

= 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

= - 4 + 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2}

- 4 + 2 π^{2} u^{3} - π^{2} u^{4} - 2 π^{2} u + π^{2} = 0

Write in the standard form

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

- π^{2} u^{4} + 2 π^{2} u^{3} - 2 π^{2} u - 4 + π^{2} = 0

Find one solution for

- 9.86960 \dots u^{4} + 19.73920 \dots u^{3} - 19.73920 \dots u + 5.86960 \dots = 0

using Newton-Raphson:

u \approx 0.32647 \dots

- 9.86960 \dots u^{4} + 19.73920 \dots u^{3} - 19.73920 \dots u + 5.86960 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 9.86960 \dots u^{4} + 19.73920 \dots u^{3} - 19.73920 \dots u + 5.86960 \dots

Find

f^{^{'}} (u) : - 39.47841 \dots u^{3} + 59.21762 \dots u^{2} - 19.73920 \dots

\frac{d}{d u} (- 9.86960 \dots u^{4} + 19.73920 \dots u^{3} - 19.73920 \dots u + 5.86960 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (9.86960 \dots u^{4}) + \frac{d}{d u} (19.73920 \dots u^{3}) - \frac{d}{d u} (19.73920 \dots u) + \frac{d}{d u} (5.86960 \dots)

\frac{d}{d u} (9.86960 \dots u^{4}) = 39.47841 \dots u^{3}

\frac{d}{d u} (9.86960 \dots u^{4})

Take the constant out:

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

= 9.86960 \dots \frac{d}{d u} (u^{4})

Apply the Power Rule:

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

= 9.86960 \dots \cdot 4 u^{4 - 1}

Simplify

= 39.47841 \dots u^{3}

\frac{d}{d u} (19.73920 \dots u^{3}) = 59.21762 \dots u^{2}

\frac{d}{d u} (19.73920 \dots u^{3})

Take the constant out:

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

= 19.73920 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

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

= 19.73920 \dots \cdot 3 u^{3 - 1}

Simplify

= 59.21762 \dots u^{2}

\frac{d}{d u} (19.73920 \dots u) = 19.73920 \dots

\frac{d}{d u} (19.73920 \dots u)

Take the constant out:

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

= 19.73920 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 19.73920 \dots \cdot 1

Simplify

= 19.73920 \dots

\frac{d}{d u} (5.86960 \dots) = 0

\frac{d}{d u} (5.86960 \dots)

Derivative of a constant:

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

= 0

= - 39.47841 \dots u^{3} + 59.21762 \dots u^{2} - 19.73920 \dots + 0

Simplify

= - 39.47841 \dots u^{3} + 59.21762 \dots u^{2} - 19.73920 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.29735 \dots : Δ u_{1} = 0.29735 \dots

f (u_{0}) = - 9.86960 \dots \cdot 0^{4} + 19.73920 \dots \cdot 0^{3} - 19.73920 \dots \cdot 0 + 5.86960 \dots = 5.86960 \dots

f^{^{'}} (u_{0}) = - 39.47841 \dots \cdot 0^{3} + 59.21762 \dots \cdot 0^{2} - 19.73920 \dots = - 19.73920 \dots

u_{1} = 0.29735 \dots

Δ u_{1} = ∣ 0.29735 \dots - 0 ∣ = 0.29735 \dots

Δ u_{1} = 0.29735 \dots

u_{2} = 0.32578 \dots : Δ u_{2} = 0.02843 \dots

f (u_{1}) = - 9.86960 \dots \cdot 0.29735 \dots^{4} + 19.73920 \dots \cdot 0.29735 \dots^{3} - 19.73920 \dots \cdot 0.29735 \dots + 5.86960 \dots = 0.44183 \dots

f^{^{'}} (u_{1}) = - 39.47841 \dots \cdot 0.29735 \dots^{3} + 59.21762 \dots \cdot 0.29735 \dots^{2} - 19.73920 \dots = - 15.54109 \dots

u_{2} = 0.32578 \dots

Δ u_{2} = ∣ 0.32578 \dots - 0.29735 \dots ∣ = 0.02843 \dots

Δ u_{2} = 0.02843 \dots

u_{3} = 0.32647 \dots : Δ u_{3} = 0.00068 \dots

f (u_{2}) = - 9.86960 \dots \cdot 0.32578 \dots^{4} + 19.73920 \dots \cdot 0.32578 \dots^{3} - 19.73920 \dots \cdot 0.32578 \dots + 5.86960 \dots = 0.01017 \dots

f^{^{'}} (u_{2}) = - 39.47841 \dots \cdot 0.32578 \dots^{3} + 59.21762 \dots \cdot 0.32578 \dots^{2} - 19.73920 \dots = - 14.81908 \dots

u_{3} = 0.32647 \dots

Δ u_{3} = ∣ 0.32647 \dots - 0.32578 \dots ∣ = 0.00068 \dots

Δ u_{3} = 0.00068 \dots

u_{4} = 0.32647 \dots : Δ u_{4} = 4.14683 E - 7

f (u_{3}) = - 9.86960 \dots \cdot 0.32647 \dots^{4} + 19.73920 \dots \cdot 0.32647 \dots^{3} - 19.73920 \dots \cdot 0.32647 \dots + 5.86960 \dots = 6.13781 E - 6

f^{^{'}} (u_{3}) = - 39.47841 \dots \cdot 0.32647 \dots^{3} + 59.21762 \dots \cdot 0.32647 \dots^{2} - 19.73920 \dots = - 14.80121 \dots

u_{4} = 0.32647 \dots

Δ u_{4} = ∣ 0.32647 \dots - 0.32647 \dots ∣ = 4.14683 E - 7

Δ u_{4} = 4.14683 E - 7

u \approx 0.32647 \dots

Apply long division:

\frac{- π ^{2} u ^{4} + 2 π ^{2} u ^{3} - 2 π ^{2} u - 4 + π ^{2}}{u - 0.32647 \dots} = - 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots

- 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots \approx 0

Find one solution for

- 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots = 0

using Newton-Raphson:

u \approx - 0.94491 \dots

- 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots

Find

f^{^{'}} (u) : - 29.60881 \dots u^{2} + 33.03405 \dots u + 5.39239 \dots

\frac{d}{d u} (- 9.86960 \dots u^{3} + 16.51702 \dots u^{2} + 5.39239 \dots u - 17.97872 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (9.86960 \dots u^{3}) + \frac{d}{d u} (16.51702 \dots u^{2}) + \frac{d}{d u} (5.39239 \dots u) - \frac{d}{d u} (17.97872 \dots)

\frac{d}{d u} (9.86960 \dots u^{3}) = 29.60881 \dots u^{2}

\frac{d}{d u} (9.86960 \dots u^{3})

Take the constant out:

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

= 9.86960 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

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

= 9.86960 \dots \cdot 3 u^{3 - 1}

Simplify

= 29.60881 \dots u^{2}

\frac{d}{d u} (16.51702 \dots u^{2}) = 33.03405 \dots u

\frac{d}{d u} (16.51702 \dots u^{2})

Take the constant out:

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

= 16.51702 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 16.51702 \dots \cdot 2 u^{2 - 1}

Simplify

= 33.03405 \dots u

\frac{d}{d u} (5.39239 \dots u) = 5.39239 \dots

\frac{d}{d u} (5.39239 \dots u)

Take the constant out:

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

= 5.39239 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 5.39239 \dots \cdot 1

Simplify

= 5.39239 \dots

\frac{d}{d u} (17.97872 \dots) = 0

\frac{d}{d u} (17.97872 \dots)

Derivative of a constant:

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

= 0

= - 29.60881 \dots u^{2} + 33.03405 \dots u + 5.39239 \dots - 0

Simplify

= - 29.60881 \dots u^{2} + 33.03405 \dots u + 5.39239 \dots

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.94732 \dots : Δ u_{1} = 0.05267 \dots

f (u_{0}) = - 9.86960 \dots (- 1)^{3} + 16.51702 \dots (- 1)^{2} + 5.39239 \dots (- 1) - 17.97872 \dots = 3.01551 \dots

f^{^{'}} (u_{0}) = - 29.60881 \dots (- 1)^{2} + 33.03405 \dots (- 1) + 5.39239 \dots = - 57.25047 \dots

u_{1} = - 0.94732 \dots

Δ u_{1} = ∣ - 0.94732 \dots - (- 1) ∣ = 0.05267 \dots

Δ u_{1} = 0.05267 \dots

u_{2} = - 0.94491 \dots : Δ u_{2} = 0.00241 \dots

f (u_{1}) = - 9.86960 \dots (- 0.94732 \dots)^{3} + 16.51702 \dots (- 0.94732 \dots)^{2} + 5.39239 \dots (- 0.94732 \dots) - 17.97872 \dots = 0.12652 \dots

f^{^{'}} (u_{1}) = - 29.60881 \dots (- 0.94732 \dots)^{2} + 33.03405 \dots (- 0.94732 \dots) + 5.39239 \dots = - 52.47351 \dots

u_{2} = - 0.94491 \dots

Δ u_{2} = ∣ - 0.94491 \dots - (- 0.94732 \dots) ∣ = 0.00241 \dots

Δ u_{2} = 0.00241 \dots

u_{3} = - 0.94491 \dots : Δ u_{3} = 4.95571 E - 6

f (u_{2}) = - 9.86960 \dots (- 0.94491 \dots)^{3} + 16.51702 \dots (- 0.94491 \dots)^{2} + 5.39239 \dots (- 0.94491 \dots) - 17.97872 \dots = 0.00025 \dots

f^{^{'}} (u_{2}) = - 29.60881 \dots (- 0.94491 \dots)^{2} + 33.03405 \dots (- 0.94491 \dots) + 5.39239 \dots = - 52.25876 \dots

u_{3} = - 0.94491 \dots

Δ u_{3} = ∣ - 0.94491 \dots - (- 0.94491 \dots) ∣ = 4.95571 E - 6

Δ u_{3} = 4.95571 E - 6

u_{4} = - 0.94491 \dots : Δ u_{4} = 2.09107 E - 11

f (u_{3}) = - 9.86960 \dots (- 0.94491 \dots)^{3} + 16.51702 \dots (- 0.94491 \dots)^{2} + 5.39239 \dots (- 0.94491 \dots) - 17.97872 \dots = 1.09276 E - 9

f^{^{'}} (u_{3}) = - 29.60881 \dots (- 0.94491 \dots)^{2} + 33.03405 \dots (- 0.94491 \dots) + 5.39239 \dots = - 52.25832 \dots

u_{4} = - 0.94491 \dots

Δ u_{4} = ∣ - 0.94491 \dots - (- 0.94491 \dots) ∣ = 2.09107 E - 11

Δ u_{4} = 2.09107 E - 11

u \approx - 0.94491 \dots

Apply long division:

\frac{- 9.86960 \dots u ^{3} + 16.51702 \dots u ^{2} + 5.39239 \dots u - 17.97872 \dots}{u + 0.94491 \dots} = - 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots

- 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots \approx 0

Find one solution for

- 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots

Find

f^{^{'}} (u) : - 19.73920 \dots u + 25.84293 \dots

\frac{d}{d u} (- 9.86960 \dots u^{2} + 25.84293 \dots u - 19.02688 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (9.86960 \dots u^{2}) + \frac{d}{d u} (25.84293 \dots u) - \frac{d}{d u} (19.02688 \dots)

\frac{d}{d u} (9.86960 \dots u^{2}) = 19.73920 \dots u

\frac{d}{d u} (9.86960 \dots u^{2})

Take the constant out:

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

= 9.86960 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 9.86960 \dots \cdot 2 u^{2 - 1}

Simplify

= 19.73920 \dots u

\frac{d}{d u} (25.84293 \dots u) = 25.84293 \dots

\frac{d}{d u} (25.84293 \dots u)

Take the constant out:

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

= 25.84293 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 25.84293 \dots \cdot 1

Simplify

= 25.84293 \dots

\frac{d}{d u} (19.02688 \dots) = 0

\frac{d}{d u} (19.02688 \dots)

Derivative of a constant:

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

= 0

= - 19.73920 \dots u + 25.84293 \dots - 0

Simplify

= - 19.73920 \dots u + 25.84293 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.50027 \dots : Δ u_{1} = 0.50027 \dots

f (u_{0}) = - 9.86960 \dots \cdot 1^{2} + 25.84293 \dots \cdot 1 - 19.02688 \dots = - 3.05355 \dots

f^{^{'}} (u_{0}) = - 19.73920 \dots \cdot 1 + 25.84293 \dots = 6.10372 \dots

u_{1} = 1.50027 \dots

Δ u_{1} = ∣ 1.50027 \dots - 1 ∣ = 0.50027 \dots

Δ u_{1} = 0.50027 \dots

u_{2} = 0.84530 \dots : Δ u_{2} = 0.65497 \dots

f (u_{1}) = - 9.86960 \dots \cdot 1.50027 \dots^{2} + 25.84293 \dots \cdot 1.50027 \dots - 19.02688 \dots = - 2.47014 \dots

f^{^{'}} (u_{1}) = - 19.73920 \dots \cdot 1.50027 \dots + 25.84293 \dots = - 3.77137 \dots

u_{2} = 0.84530 \dots

Δ u_{2} = ∣ 0.84530 \dots - 1.50027 \dots ∣ = 0.65497 \dots

Δ u_{2} = 0.65497 \dots

u_{3} = 1.30766 \dots : Δ u_{3} = 0.46235 \dots

f (u_{2}) = - 9.86960 \dots \cdot 0.84530 \dots^{2} + 25.84293 \dots \cdot 0.84530 \dots - 19.02688 \dots = - 4.23396 \dots

f^{^{'}} (u_{2}) = - 19.73920 \dots \cdot 0.84530 \dots + 25.84293 \dots = 9.15728 \dots

u_{3} = 1.30766 \dots

Δ u_{3} = ∣ 1.30766 \dots - 0.84530 \dots ∣ = 0.46235 \dots

Δ u_{3} = 0.46235 \dots

u_{4} = 70.11579 \dots : Δ u_{4} = 68.80812 \dots

f (u_{3}) = - 9.86960 \dots \cdot 1.30766 \dots^{2} + 25.84293 \dots \cdot 1.30766 \dots - 19.02688 \dots = - 2.10989 \dots

f^{^{'}} (u_{3}) = - 19.73920 \dots \cdot 1.30766 \dots + 25.84293 \dots = 0.03066 \dots

u_{4} = 70.11579 \dots

Δ u_{4} = ∣ 70.11579 \dots - 1.30766 \dots ∣ = 68.80812 \dots

Δ u_{4} = 68.80812 \dots

u_{5} = 35.71095 \dots : Δ u_{5} = 34.40483 \dots

f (u_{4}) = - 9.86960 \dots \cdot 70.11579 \dots^{2} + 25.84293 \dots \cdot 70.11579 \dots - 19.02688 \dots = - 46728.21617 \dots

f^{^{'}} (u_{4}) = - 19.73920 \dots \cdot 70.11579 \dots + 25.84293 \dots = - 1358.18729 \dots

u_{5} = 35.71095 \dots

Δ u_{5} = ∣ 35.71095 \dots - 70.11579 \dots ∣ = 34.40483 \dots

Δ u_{5} = 34.40483 \dots

u_{6} = 18.50697 \dots : Δ u_{6} = 17.20397 \dots

f (u_{5}) = - 9.86960 \dots \cdot 35.71095 \dots^{2} + 25.84293 \dots \cdot 35.71095 \dots - 19.02688 \dots = - 11682.58153 \dots

f^{^{'}} (u_{5}) = - 19.73920 \dots \cdot 35.71095 \dots + 25.84293 \dots = - 679.06298 \dots

u_{6} = 18.50697 \dots

Δ u_{6} = ∣ 18.50697 \dots - 35.71095 \dots ∣ = 17.20397 \dots

Δ u_{6} = 17.20397 \dots

u_{7} = 9.90188 \dots : Δ u_{7} = 8.60509 \dots

f (u_{6}) = - 9.86960 \dots \cdot 18.50697 \dots^{2} + 25.84293 \dots \cdot 18.50697 \dots - 19.02688 \dots = - 2921.17294 \dots

f^{^{'}} (u_{6}) = - 19.73920 \dots \cdot 18.50697 \dots + 25.84293 \dots = - 339.47016 \dots

u_{7} = 9.90188 \dots

Δ u_{7} = ∣ 9.90188 \dots - 18.50697 \dots ∣ = 8.60509 \dots

Δ u_{7} = 8.60509 \dots

u_{8} = 5.59311 \dots : Δ u_{8} = 4.30877 \dots

f (u_{7}) = - 9.86960 \dots \cdot 9.90188 \dots^{2} + 25.84293 \dots \cdot 9.90188 \dots - 19.02688 \dots = - 730.82108 \dots

f^{^{'}} (u_{7}) = - 19.73920 \dots \cdot 9.90188 \dots + 25.84293 \dots = - 169.61239 \dots

u_{8} = 5.59311 \dots

Δ u_{8} = ∣ 5.59311 \dots - 9.90188 \dots ∣ = 4.30877 \dots

Δ u_{8} = 4.30877 \dots

u_{9} = 3.42621 \dots : Δ u_{9} = 2.16689 \dots

f (u_{8}) = - 9.86960 \dots \cdot 5.59311 \dots^{2} + 25.84293 \dots \cdot 5.59311 \dots - 19.02688 \dots = - 183.23426 \dots

f^{^{'}} (u_{8}) = - 19.73920 \dots \cdot 5.59311 \dots + 25.84293 \dots = - 84.56065 \dots

u_{9} = 3.42621 \dots

Δ u_{9} = ∣ 3.42621 \dots - 5.59311 \dots ∣ = 2.16689 \dots

Δ u_{9} = 2.16689 \dots

u_{10} = 2.31722 \dots : Δ u_{10} = 1.10898 \dots

f (u_{9}) = - 9.86960 \dots \cdot 3.42621 \dots^{2} + 25.84293 \dots \cdot 3.42621 \dots - 19.02688 \dots = - 46.34217 \dots

f^{^{'}} (u_{9}) = - 19.73920 \dots \cdot 3.42621 \dots + 25.84293 \dots = - 41.78781 \dots

u_{10} = 2.31722 \dots

Δ u_{10} = ∣ 2.31722 \dots - 3.42621 \dots ∣ = 1.10898 \dots

Δ u_{10} = 1.10898 \dots

u_{11} = 1.70718 \dots : Δ u_{11} = 0.61004 \dots

f (u_{10}) = - 9.86960 \dots \cdot 2.31722 \dots^{2} + 25.84293 \dots \cdot 2.31722 \dots - 19.02688 \dots = - 12.13817 \dots

f^{^{'}} (u_{10}) = - 19.73920 \dots \cdot 2.31722 \dots + 25.84293 \dots = - 19.89727 \dots

u_{11} = 1.70718 \dots

Δ u_{11} = ∣ 1.70718 \dots - 2.31722 \dots ∣ = 0.61004 \dots

Δ u_{11} = 0.61004 \dots

u_{12} = 1.23961 \dots : Δ u_{12} = 0.46756 \dots

f (u_{11}) = - 9.86960 \dots \cdot 1.70718 \dots^{2} + 25.84293 \dots \cdot 1.70718 \dots - 19.02688 \dots = - 3.67298 \dots

f^{^{'}} (u_{11}) = - 19.73920 \dots \cdot 1.70718 \dots + 25.84293 \dots = - 7.85553 \dots

u_{12} = 1.23961 \dots

Δ u_{12} = ∣ 1.23961 \dots - 1.70718 \dots ∣ = 0.46756 \dots

Δ u_{12} = 0.46756 \dots

u_{13} = 2.81013 \dots : Δ u_{13} = 1.57051 \dots

f (u_{12}) = - 9.86960 \dots \cdot 1.23961 \dots^{2} + 25.84293 \dots \cdot 1.23961 \dots - 19.02688 \dots = - 2.15767 \dots

f^{^{'}} (u_{12}) = - 19.73920 \dots \cdot 1.23961 \dots + 25.84293 \dots = 1.37386 \dots

u_{13} = 2.81013 \dots

Δ u_{13} = ∣ 2.81013 \dots - 1.23961 \dots ∣ = 1.57051 \dots

Δ u_{13} = 1.57051 \dots

u_{14} = 1.98846 \dots : Δ u_{14} = 0.82167 \dots

f (u_{13}) = - 9.86960 \dots \cdot 2.81013 \dots^{2} + 25.84293 \dots \cdot 2.81013 \dots - 19.02688 \dots = - 24.34357 \dots

f^{^{'}} (u_{13}) = - 19.73920 \dots \cdot 2.81013 \dots + 25.84293 \dots = - 29.62687 \dots

u_{14} = 1.98846 \dots

Δ u_{14} = ∣ 1.98846 \dots - 2.81013 \dots ∣ = 0.82167 \dots

Δ u_{14} = 0.82167 \dots

u_{15} = 1.49147 \dots : Δ u_{15} = 0.49698 \dots

f (u_{14}) = - 9.86960 \dots \cdot 1.98846 \dots^{2} + 25.84293 \dots \cdot 1.98846 \dots - 19.02688 \dots = - 6.66341 \dots

f^{^{'}} (u_{14}) = - 19.73920 \dots \cdot 1.98846 \dots + 25.84293 \dots = - 13.40771 \dots

u_{15} = 1.49147 \dots

Δ u_{15} = ∣ 1.49147 \dots - 1.98846 \dots ∣ = 0.49698 \dots

Δ u_{15} = 0.49698 \dots

u_{16} = 0.81389 \dots : Δ u_{16} = 0.67758 \dots

f (u_{15}) = - 9.86960 \dots \cdot 1.49147 \dots^{2} + 25.84293 \dots \cdot 1.49147 \dots - 19.02688 \dots = - 2.43772 \dots

f^{^{'}} (u_{15}) = - 19.73920 \dots \cdot 1.49147 \dots + 25.84293 \dots = - 3.59765 \dots

u_{16} = 0.81389 \dots

Δ u_{16} = ∣ 0.81389 \dots - 1.49147 \dots ∣ = 0.67758 \dots

Δ u_{16} = 0.67758 \dots

Cannot find solution

The solutions are

u \approx 0.32647 \dots, u \approx - 0.94491 \dots

Substitute back

u = cos (c)

cos (c) \approx 0.32647 \dots, cos (c) \approx - 0.94491 \dots

cos (c) \approx 0.32647 \dots, cos (c) \approx - 0.94491 \dots

cos (c) = 0.32647 \dots : c = arccos (0.32647 \dots) + 2 πn, c = 2 π - arccos (0.32647 \dots) + 2 πn

cos (c) = 0.32647 \dots

Apply trig inverse properties

cos (c) = 0.32647 \dots

General solutions for

cos (c) = 0.32647 \dots

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

c = arccos (0.32647 \dots) + 2 πn, c = 2 π - arccos (0.32647 \dots) + 2 πn

c = arccos (0.32647 \dots) + 2 πn, c = 2 π - arccos (0.32647 \dots) + 2 πn

cos (c) = - 0.94491 \dots : c = arccos (- 0.94491 \dots) + 2 πn, c = - arccos (- 0.94491 \dots) + 2 πn

cos (c) = - 0.94491 \dots

Apply trig inverse properties

cos (c) = - 0.94491 \dots

General solutions for

cos (c) = - 0.94491 \dots

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

c = arccos (- 0.94491 \dots) + 2 πn, c = - arccos (- 0.94491 \dots) + 2 πn

c = arccos (- 0.94491 \dots) + 2 πn, c = - arccos (- 0.94491 \dots) + 2 πn

Combine all the solutions

c = arccos (0.32647 \dots) + 2 πn, c = 2 π - arccos (0.32647 \dots) + 2 πn, c = arccos (- 0.94491 \dots) + 2 πn, c = - arccos (- 0.94491 \dots) + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

Remove the ones that don't agree with the equation.

Check the solution

arccos (0.32647 \dots) + 2 πn :

True

arccos (0.32647 \dots) + 2 πn

Plug in

n = 1

arccos (0.32647 \dots) + 2 π 1

For

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

plug in

c = arccos (0.32647 \dots) + 2 π 1

sin (arccos (0.32647 \dots) + 2 π 1) = \frac{2}{π ( - cos ( arccos ( 0.32647 \dots ) + 2 π 1 ) + 1 )}

Refine

0.94520 \dots = 0.94520 \dots

\Rightarrow True

Check the solution

2 π - arccos (0.32647 \dots) + 2 πn :

False

2 π - arccos (0.32647 \dots) + 2 πn

Plug in

n = 1

2 π - arccos (0.32647 \dots) + 2 π 1

For

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

plug in

c = 2 π - arccos (0.32647 \dots) + 2 π 1

sin (2 π - arccos (0.32647 \dots) + 2 π 1) = \frac{2}{π ( - cos ( 2 π - arccos ( 0.32647 \dots ) + 2 π 1 ) + 1 )}

Refine

- 0.94520 \dots = 0.94520 \dots

\Rightarrow False

Check the solution

arccos (- 0.94491 \dots) + 2 πn :

True

arccos (- 0.94491 \dots) + 2 πn

Plug in

n = 1

arccos (- 0.94491 \dots) + 2 π 1

For

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

plug in

c = arccos (- 0.94491 \dots) + 2 π 1

sin (arccos (- 0.94491 \dots) + 2 π 1) = \frac{2}{π ( - cos ( arccos ( - 0.94491 \dots ) + 2 π 1 ) + 1 )}

Refine

0.32732 \dots = 0.32732 \dots

\Rightarrow True

Check the solution

- arccos (- 0.94491 \dots) + 2 πn :

False

- arccos (- 0.94491 \dots) + 2 πn

Plug in

n = 1

- arccos (- 0.94491 \dots) + 2 π 1

For

sin (c) = \frac{2}{π ( - cos ( c ) + 1 )}

plug in

c = - arccos (- 0.94491 \dots) + 2 π 1

sin (- arccos (- 0.94491 \dots) + 2 π 1) = \frac{2}{π ( - cos ( - arccos ( - 0.94491 \dots ) + 2 π 1 ) + 1 )}

Refine

- 0.32732 \dots = 0.32732 \dots

\Rightarrow False

c = arccos (0.32647 \dots) + 2 πn, c = arccos (- 0.94491 \dots) + 2 πn

Show solutions in decimal form

c = 1.23822 \dots + 2 πn, c = 2.80812 \dots + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin(c)= 2/(pi(-cos(c)+1)) ?
The general solution for sin(c)= 2/(pi(-cos(c)+1)) is c=1.23822…+2pin,c=2.80812…+2pin