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Popular Trigonometry >

sqrt(1-cos(x))= 1/(2sin^2(x))

Solution

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

Solution

x = 1.01879 \dots + 2 πn, x = 2 π - 1.01879 \dots + 2 πn, x = 2.48401 \dots + 2 πn, x = - 2.48401 \dots + 2 πn

Degrees

x = 58.37265 \dots^{\circ} + 36 0^{\circ} n, x = 301.62734 \dots^{\circ} + 36 0^{\circ} n, x = 142.32379 \dots^{\circ} + 36 0^{\circ} n, x = - 142.32379 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

\frac{1}{2 sin ^{2} ( x )}

from both sides

1 - cos (x) - \frac{1}{2 sin ^{2} ( x )} = 0

Simplify

1 - cos (x) - \frac{1}{2 sin ^{2} ( x )} : \frac{2 sin ^{2} ( x ) 1 - cos ( x ) - 1}{2 sin ^{2} ( x )}

1 - cos (x) - \frac{1}{2 sin ^{2} ( x )}

Convert element to fraction:

- cos (x) + 1 = \frac{1 - cos ( x ) \cdot 2 sin ^{2} ( x )}{2 sin ^{2} ( x )}

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

Since the denominators are equal, combine the fractions:

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

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

\frac{2 sin ^{2} ( x ) 1 - cos ( x ) - 1}{2 sin ^{2} ( x )} = 0

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

2 sin^{2} (x) 1 - cos (x) - 1 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

- 1 + (1 - cos^{2} (x)) \cdot 2 1 - cos (x) = 0

Solve by substitution

- 1 + (1 - cos^{2} (x)) \cdot 2 1 - cos (x) = 0

Let:

cos (x) = u

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

- 1 + (1 - u^{2}) \cdot 2 1 - u = 0 : u \approx 0.52439 \dots, u \approx - 0.79147 \dots

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

Expand

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 2 1 - u, b = 1, c = u^{2}

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

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

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

Factor

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

2 1 - u - 2 1 - u u^{2}

Rewrite as

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

Factor out common term

2 1 - u

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

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

Square both sides:

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1

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

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

Expand

(2 1 - u (1 - u^{2}))^{2} : 4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5}

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

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

Apply radical rule:

a = a^{\frac{1}{2}}

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 1 - u

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

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

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

Apply Perfect Square Formula:

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

a = 1, b = u^{2}

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

2 \cdot 1 \cdot u^{2}

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

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

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

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

2^{2} = 4

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

Simplify

4 \cdot 1 \cdot 1 - u - 4 \cdot 21 - u u^{2} + 41 - u u^{4} : 41 - u - 81 - u u^{2} + 41 - u u^{4}

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

Multiply the numbers:

4 \cdot 1 = 4

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

Multiply the numbers:

4 \cdot 2 = 8

= 4 (1 - u) - 8 (1 - u) u^{2} + 4 (1 - u) u^{4}

= 4 (1 - u) - 8 (1 - u) u^{2} + 4 (1 - u) u^{4}

Expand

4 (1 - u) - 8 (1 - u) u^{2} + 4 (1 - u) u^{4} : 4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5}

4 (1 - u) - 8 (1 - u) u^{2} + 4 (1 - u) u^{4}

= 4 (1 - u) - 8 u^{2} (1 - u) + 4 u^{4} (1 - u)

Expand

4 (1 - u) : 4 - 4 u

4 (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = u

= 4 \cdot 1 - 4 u

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 u

= 4 - 4 u - 8 (1 - u) u^{2} + 4 (1 - u) u^{4}

Expand

- 8 u^{2} (1 - u) : - 8 u^{2} + 8 u^{3}

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 8 u^{2}, b = 1, c = u

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

Apply minus-plus rules

- (- a) = a

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

Simplify

- 8 \cdot 1 \cdot u^{2} + 8 u^{2} u : - 8 u^{2} + 8 u^{3}

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

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

8 \cdot 1 \cdot u^{2}

Multiply the numbers:

8 \cdot 1 = 8

= 8 u^{2}

8 u^{2} u = 8 u^{3}

8 u^{2} u

Apply exponent rule:

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

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

= 8 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 8 u^{3}

= - 8 u^{2} + 8 u^{3}

= - 8 u^{2} + 8 u^{3}

= 4 - 4 u - 8 u^{2} + 8 u^{3} + 4 (1 - u) u^{4}

Expand

4 u^{4} (1 - u) : 4 u^{4} - 4 u^{5}

4 u^{4} (1 - u)

Apply the distributive law:

a (b - c) = ab - a c

a = 4 u^{4}, b = 1, c = u

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

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

Simplify

4 \cdot 1 \cdot u^{4} - 4 u^{4} u : 4 u^{4} - 4 u^{5}

4 \cdot 1 \cdot u^{4} - 4 u^{4} u

4 \cdot 1 \cdot u^{4} = 4 u^{4}

4 \cdot 1 \cdot u^{4}

Multiply the numbers:

4 \cdot 1 = 4

= 4 u^{4}

4 u^{4} u = 4 u^{5}

4 u^{4} u

Apply exponent rule:

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

u^{4} u = u^{4 + 1}

= 4 u^{4 + 1}

Add the numbers:

4 + 1 = 5

= 4 u^{5}

= 4 u^{4} - 4 u^{5}

= 4 u^{4} - 4 u^{5}

= 4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5}

= 4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5}

Expand

1^{2} : 1

1^{2}

Apply rule

1^{a} = 1

= 1

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1

Solve

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1 : u \approx - 1.15774 \dots, u \approx 0.52439 \dots, u \approx - 0.79147 \dots

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1

Move

1

to the left side

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} = 1

Subtract

1

from both sides

4 - 4 u - 8 u^{2} + 8 u^{3} + 4 u^{4} - 4 u^{5} - 1 = 1 - 1

Simplify

- 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3 = 0

- 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3 = 0

Find one solution for

- 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3 = 0

using Newton-Raphson:

u \approx - 1.15774 \dots

- 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3 = 0

Newton-Raphson Approximation Definition

f (u) = - 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3

Find

f^{^{'}} (u) : - 20 u^{4} + 16 u^{3} + 24 u^{2} - 16 u - 4

\frac{d}{d u} (- 4 u^{5} + 4 u^{4} + 8 u^{3} - 8 u^{2} - 4 u + 3)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (4 u^{5}) + \frac{d}{d u} (4 u^{4}) + \frac{d}{d u} (8 u^{3}) - \frac{d}{d u} (8 u^{2}) - \frac{d}{d u} (4 u) + \frac{d}{d u} (3)

\frac{d}{d u} (4 u^{5}) = 20 u^{4}

\frac{d}{d u} (4 u^{5})

Take the constant out:

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

= 4 \frac{d}{d u} (u^{5})

Apply the Power Rule:

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

= 4 \cdot 5 u^{5 - 1}

Simplify

= 20 u^{4}

\frac{d}{d u} (4 u^{4}) = 16 u^{3}

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

Take the constant out:

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

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

Apply the Power Rule:

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

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

Simplify

= 16 u^{3}

\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} (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} (4 u) = 4

\frac{d}{d u} (4 u)

Take the constant out:

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

= 4 \frac{d u}{d u}

Apply the common derivative:

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

= 4 \cdot 1

Simplify

= 4

\frac{d}{d u} (3) = 0

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

Derivative of a constant:

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

= 0

= - 20 u^{4} + 16 u^{3} + 24 u^{2} - 16 u - 4 + 0

Simplify

= - 20 u^{4} + 16 u^{3} + 24 u^{2} - 16 u - 4

Let

u_{0} = 2

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.71969 \dots : Δ u_{1} = 0.28030 \dots

f (u_{0}) = - 4 \cdot 2^{5} + 4 \cdot 2^{4} + 8 \cdot 2^{3} - 8 \cdot 2^{2} - 4 \cdot 2 + 3 = - 37

f^{^{'}} (u_{0}) = - 20 \cdot 2^{4} + 16 \cdot 2^{3} + 24 \cdot 2^{2} - 16 \cdot 2 - 4 = - 132

u_{1} = 1.71969 \dots

Δ u_{1} = ∣ 1.71969 \dots - 2 ∣ = 0.28030 \dots

Δ u_{1} = 0.28030 \dots

u_{2} = 1.49728 \dots : Δ u_{2} = 0.22241 \dots

f (u_{1}) = - 4 \cdot 1.71969 \dots^{5} + 4 \cdot 1.71969 \dots^{4} + 8 \cdot 1.71969 \dots^{3} - 8 \cdot 1.71969 \dots^{2} - 4 \cdot 1.71969 \dots + 3 = - 12.02935 \dots

f^{^{'}} (u_{1}) = - 20 \cdot 1.71969 \dots^{4} + 16 \cdot 1.71969 \dots^{3} + 24 \cdot 1.71969 \dots^{2} - 16 \cdot 1.71969 \dots - 4 = - 54.08571 \dots

u_{2} = 1.49728 \dots

Δ u_{2} = ∣ 1.49728 \dots - 1.71969 \dots ∣ = 0.22241 \dots

Δ u_{2} = 0.22241 \dots

u_{3} = 1.30324 \dots : Δ u_{3} = 0.19403 \dots

f (u_{2}) = - 4 \cdot 1.49728 \dots^{5} + 4 \cdot 1.49728 \dots^{4} + 8 \cdot 1.49728 \dots^{3} - 8 \cdot 1.49728 \dots^{2} - 4 \cdot 1.49728 \dots + 3 = - 4.06767 \dots

f^{^{'}} (u_{2}) = - 20 \cdot 1.49728 \dots^{4} + 16 \cdot 1.49728 \dots^{3} + 24 \cdot 1.49728 \dots^{2} - 16 \cdot 1.49728 \dots - 4 = - 20.96340 \dots

u_{3} = 1.30324 \dots

Δ u_{3} = ∣ 1.30324 \dots - 1.49728 \dots ∣ = 0.19403 \dots

Δ u_{3} = 0.19403 \dots

u_{4} = 1.05328 \dots : Δ u_{4} = 0.24996 \dots

f (u_{3}) = - 4 \cdot 1.30324 \dots^{5} + 4 \cdot 1.30324 \dots^{4} + 8 \cdot 1.30324 \dots^{3} - 8 \cdot 1.30324 \dots^{2} - 4 \cdot 1.30324 \dots + 3 = - 1.59173 \dots

f^{^{'}} (u_{3}) = - 20 \cdot 1.30324 \dots^{4} + 16 \cdot 1.30324 \dots^{3} + 24 \cdot 1.30324 \dots^{2} - 16 \cdot 1.30324 \dots - 4 = - 6.36786 \dots

u_{4} = 1.05328 \dots

Δ u_{4} = ∣ 1.05328 \dots - 1.30324 \dots ∣ = 0.24996 \dots

Δ u_{4} = 0.24996 \dots

u_{5} = - 5.80799 \dots : Δ u_{5} = 6.86128 \dots

f (u_{4}) = - 4 \cdot 1.05328 \dots^{5} + 4 \cdot 1.05328 \dots^{4} + 8 \cdot 1.05328 \dots^{3} - 8 \cdot 1.05328 \dots^{2} - 4 \cdot 1.05328 \dots + 3 = - 1.00255 \dots

f^{^{'}} (u_{4}) = - 20 \cdot 1.05328 \dots^{4} + 16 \cdot 1.05328 \dots^{3} + 24 \cdot 1.05328 \dots^{2} - 16 \cdot 1.05328 \dots - 4 = - 0.14611 \dots

u_{5} = - 5.80799 \dots

Δ u_{5} = ∣ - 5.80799 \dots - 1.05328 \dots ∣ = 6.86128 \dots

Δ u_{5} = 6.86128 \dots

u_{6} = - 4.64067 \dots : Δ u_{6} = 1.16732 \dots

f (u_{5}) = - 4 (- 5.80799 \dots)^{5} + 4 (- 5.80799 \dots)^{4} + 8 (- 5.80799 \dots)^{3} - 8 (- 5.80799 \dots)^{2} - 4 (- 5.80799 \dots) + 3 = 29176.40873 \dots

f^{^{'}} (u_{5}) = - 20 (- 5.80799 \dots)^{4} + 16 (- 5.80799 \dots)^{3} + 24 (- 5.80799 \dots)^{2} - 16 (- 5.80799 \dots) - 4 = - 24994.29514 \dots

u_{6} = - 4.64067 \dots

Δ u_{6} = ∣ - 4.64067 \dots - (- 5.80799 \dots) ∣ = 1.16732 \dots

Δ u_{6} = 1.16732 \dots

u_{7} = - 3.71587 \dots : Δ u_{7} = 0.92480 \dots

f (u_{6}) = - 4 (- 4.64067 \dots)^{5} + 4 (- 4.64067 \dots)^{4} + 8 (- 4.64067 \dots)^{3} - 8 (- 4.64067 \dots)^{2} - 4 (- 4.64067 \dots) + 3 = 9514.18126 \dots

f^{^{'}} (u_{6}) = - 20 (- 4.64067 \dots)^{4} + 16 (- 4.64067 \dots)^{3} + 24 (- 4.64067 \dots)^{2} - 16 (- 4.64067 \dots) - 4 = - 10287.81312 \dots

u_{7} = - 3.71587 \dots

Δ u_{7} = ∣ - 3.71587 \dots - (- 4.64067 \dots) ∣ = 0.92480 \dots

Δ u_{7} = 0.92480 \dots

u_{8} = - 2.98754 \dots : Δ u_{8} = 0.72832 \dots

f (u_{7}) = - 4 (- 3.71587 \dots)^{5} + 4 (- 3.71587 \dots)^{4} + 8 (- 3.71587 \dots)^{3} - 8 (- 3.71587 \dots)^{2} - 4 (- 3.71587 \dots) + 3 = 3093.32373 \dots

f^{^{'}} (u_{7}) = - 20 (- 3.71587 \dots)^{4} + 16 (- 3.71587 \dots)^{3} + 24 (- 3.71587 \dots)^{2} - 16 (- 3.71587 \dots) - 4 = - 4247.14664 \dots

u_{8} = - 2.98754 \dots

Δ u_{8} = ∣ - 2.98754 \dots - (- 3.71587 \dots) ∣ = 0.72832 \dots

Δ u_{8} = 0.72832 \dots

u_{9} = - 2.41948 \dots : Δ u_{9} = 0.56806 \dots

f (u_{8}) = - 4 (- 2.98754 \dots)^{5} + 4 (- 2.98754 \dots)^{4} + 8 (- 2.98754 \dots)^{3} - 8 (- 2.98754 \dots)^{2} - 4 (- 2.98754 \dots) + 3 = 1000.86681 \dots

f^{^{'}} (u_{8}) = - 20 (- 2.98754 \dots)^{4} + 16 (- 2.98754 \dots)^{3} + 24 (- 2.98754 \dots)^{2} - 16 (- 2.98754 \dots) - 4 = - 1761.89279 \dots

u_{9} = - 2.41948 \dots

Δ u_{9} = ∣ - 2.41948 \dots - (- 2.98754 \dots) ∣ = 0.56806 \dots

Δ u_{9} = 0.56806 \dots

u_{10} = - 1.98344 \dots : Δ u_{10} = 0.43603 \dots

f (u_{9}) = - 4 (- 2.41948 \dots)^{5} + 4 (- 2.41948 \dots)^{4} + 8 (- 2.41948 \dots)^{3} - 8 (- 2.41948 \dots)^{2} - 4 (- 2.41948 \dots) + 3 = 321.25479 \dots

f^{^{'}} (u_{9}) = - 20 (- 2.41948 \dots)^{4} + 16 (- 2.41948 \dots)^{3} + 24 (- 2.41948 \dots)^{2} - 16 (- 2.41948 \dots) - 4 = - 736.76863 \dots

u_{10} = - 1.98344 \dots

Δ u_{10} = ∣ - 1.98344 \dots - (- 2.41948 \dots) ∣ = 0.43603 \dots

Δ u_{10} = 0.43603 \dots

u_{11} = - 1.65761 \dots : Δ u_{11} = 0.32583 \dots

f (u_{10}) = - 4 (- 1.98344 \dots)^{5} + 4 (- 1.98344 \dots)^{4} + 8 (- 1.98344 \dots)^{3} - 8 (- 1.98344 \dots)^{2} - 4 (- 1.98344 \dots) + 3 = 101.73511 \dots

f^{^{'}} (u_{10}) = - 20 (- 1.98344 \dots)^{4} + 16 (- 1.98344 \dots)^{3} + 24 (- 1.98344 \dots)^{2} - 16 (- 1.98344 \dots) - 4 = - 312.23335 \dots

u_{11} = - 1.65761 \dots

Δ u_{11} = ∣ - 1.65761 \dots - (- 1.98344 \dots) ∣ = 0.32583 \dots

Δ u_{11} = 0.32583 \dots

u_{12} = - 1.42520 \dots : Δ u_{12} = 0.23241 \dots

f (u_{11}) = - 4 (- 1.65761 \dots)^{5} + 4 (- 1.65761 \dots)^{4} + 8 (- 1.65761 \dots)^{3} - 8 (- 1.65761 \dots)^{2} - 4 (- 1.65761 \dots) + 3 = 31.47022 \dots

f^{^{'}} (u_{11}) = - 20 (- 1.65761 \dots)^{4} + 16 (- 1.65761 \dots)^{3} + 24 (- 1.65761 \dots)^{2} - 16 (- 1.65761 \dots) - 4 = - 135.40437 \dots

u_{12} = - 1.42520 \dots

Δ u_{12} = ∣ - 1.42520 \dots - (- 1.65761 \dots) ∣ = 0.23241 \dots

Δ u_{12} = 0.23241 \dots

u_{13} = - 1.27318 \dots : Δ u_{13} = 0.15201 \dots

f (u_{12}) = - 4 (- 1.42520 \dots)^{5} + 4 (- 1.42520 \dots)^{4} + 8 (- 1.42520 \dots)^{3} - 8 (- 1.42520 \dots)^{2} - 4 (- 1.42520 \dots) + 3 = 9.31556 \dots

f^{^{'}} (u_{12}) = - 20 (- 1.42520 \dots)^{4} + 16 (- 1.42520 \dots)^{3} + 24 (- 1.42520 \dots)^{2} - 16 (- 1.42520 \dots) - 4 = - 61.28130 \dots

u_{13} = - 1.27318 \dots

Δ u_{13} = ∣ - 1.27318 \dots - (- 1.42520 \dots) ∣ = 0.15201 \dots

Δ u_{13} = 0.15201 \dots

u_{14} = - 1.19046 \dots : Δ u_{14} = 0.08272 \dots

f (u_{13}) = - 4 (- 1.27318 \dots)^{5} + 4 (- 1.27318 \dots)^{4} + 8 (- 1.27318 \dots)^{3} - 8 (- 1.27318 \dots)^{2} - 4 (- 1.27318 \dots) + 3 = 2.50663 \dots

f^{^{'}} (u_{13}) = - 20 (- 1.27318 \dots)^{4} + 16 (- 1.27318 \dots)^{3} + 24 (- 1.27318 \dots)^{2} - 16 (- 1.27318 \dots) - 4 = - 30.29974 \dots

u_{14} = - 1.19046 \dots

Δ u_{14} = ∣ - 1.19046 \dots - (- 1.27318 \dots) ∣ = 0.08272 \dots

Δ u_{14} = 0.08272 \dots

u_{15} = - 1.16145 \dots : Δ u_{15} = 0.02900 \dots

f (u_{14}) = - 4 (- 1.19046 \dots)^{5} + 4 (- 1.19046 \dots)^{4} + 8 (- 1.19046 \dots)^{3} - 8 (- 1.19046 \dots)^{2} - 4 (- 1.19046 \dots) + 3 = 0.52502 \dots

f^{^{'}} (u_{14}) = - 20 (- 1.19046 \dots)^{4} + 16 (- 1.19046 \dots)^{3} + 24 (- 1.19046 \dots)^{2} - 16 (- 1.19046 \dots) - 4 = - 18.10266 \dots

u_{15} = - 1.16145 \dots

Δ u_{15} = ∣ - 1.16145 \dots - (- 1.19046 \dots) ∣ = 0.02900 \dots

Δ u_{15} = 0.02900 \dots

u_{16} = - 1.15780 \dots : Δ u_{16} = 0.00365 \dots

f (u_{15}) = - 4 (- 1.16145 \dots)^{5} + 4 (- 1.16145 \dots)^{4} + 8 (- 1.16145 \dots)^{3} - 8 (- 1.16145 \dots)^{2} - 4 (- 1.16145 \dots) + 3 = 0.05297 \dots

f^{^{'}} (u_{15}) = - 20 (- 1.16145 \dots)^{4} + 16 (- 1.16145 \dots)^{3} + 24 (- 1.16145 \dots)^{2} - 16 (- 1.16145 \dots) - 4 = - 14.50486 \dots

u_{16} = - 1.15780 \dots

Δ u_{16} = ∣ - 1.15780 \dots - (- 1.16145 \dots) ∣ = 0.00365 \dots

Δ u_{16} = 0.00365 \dots

u_{17} = - 1.15774 \dots : Δ u_{17} = 0.00005 \dots

f (u_{16}) = - 4 (- 1.15780 \dots)^{5} + 4 (- 1.15780 \dots)^{4} + 8 (- 1.15780 \dots)^{3} - 8 (- 1.15780 \dots)^{2} - 4 (- 1.15780 \dots) + 3 = 0.00078 \dots

f^{^{'}} (u_{16}) = - 20 (- 1.15780 \dots)^{4} + 16 (- 1.15780 \dots)^{3} + 24 (- 1.15780 \dots)^{2} - 16 (- 1.15780 \dots) - 4 = - 14.07518 \dots

u_{17} = - 1.15774 \dots

Δ u_{17} = ∣ - 1.15774 \dots - (- 1.15780 \dots) ∣ = 0.00005 \dots

Δ u_{17} = 0.00005 \dots

u_{18} = - 1.15774 \dots : Δ u_{18} = 1.29677 E - 8

f (u_{17}) = - 4 (- 1.15774 \dots)^{5} + 4 (- 1.15774 \dots)^{4} + 8 (- 1.15774 \dots)^{3} - 8 (- 1.15774 \dots)^{2} - 4 (- 1.15774 \dots) + 3 = 1.82438 E - 7

f^{^{'}} (u_{17}) = - 20 (- 1.15774 \dots)^{4} + 16 (- 1.15774 \dots)^{3} + 24 (- 1.15774 \dots)^{2} - 16 (- 1.15774 \dots) - 4 = - 14.06865 \dots

u_{18} = - 1.15774 \dots

Δ u_{18} = ∣ - 1.15774 \dots - (- 1.15774 \dots) ∣ = 1.29677 E - 8

Δ u_{18} = 1.29677 E - 8

u \approx - 1.15774 \dots

Apply long division:

\frac{- 4 u ^{5} + 4 u ^{4} + 8 u ^{3} - 8 u ^{2} - 4 u + 3}{u + 1.15774 \dots} = - 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots

- 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots \approx 0

Find one solution for

- 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots = 0

using Newton-Raphson:

u \approx 0.52439 \dots

- 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots

Find

f^{^{'}} (u) : - 16 u^{3} + 25.89299 \dots u^{2} - 3.98507 \dots u - 5.69314 \dots

\frac{d}{d u} (- 4 u^{4} + 8.63099 \dots u^{3} - 1.99253 \dots u^{2} - 5.69314 \dots u + 2.59123 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (4 u^{4}) + \frac{d}{d u} (8.63099 \dots u^{3}) - \frac{d}{d u} (1.99253 \dots u^{2}) - \frac{d}{d u} (5.69314 \dots u) + \frac{d}{d u} (2.59123 \dots)

\frac{d}{d u} (4 u^{4}) = 16 u^{3}

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

Take the constant out:

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

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

Apply the Power Rule:

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

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

Simplify

= 16 u^{3}

\frac{d}{d u} (8.63099 \dots u^{3}) = 25.89299 \dots u^{2}

\frac{d}{d u} (8.63099 \dots u^{3})

Take the constant out:

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

= 8.63099 \dots \frac{d}{d u} (u^{3})

Apply the Power Rule:

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

= 8.63099 \dots \cdot 3 u^{3 - 1}

Simplify

= 25.89299 \dots u^{2}

\frac{d}{d u} (1.99253 \dots u^{2}) = 3.98507 \dots u

\frac{d}{d u} (1.99253 \dots u^{2})

Take the constant out:

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

= 1.99253 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 1.99253 \dots \cdot 2 u^{2 - 1}

Simplify

= 3.98507 \dots u

\frac{d}{d u} (5.69314 \dots u) = 5.69314 \dots

\frac{d}{d u} (5.69314 \dots u)

Take the constant out:

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

= 5.69314 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 5.69314 \dots \cdot 1

Simplify

= 5.69314 \dots

\frac{d}{d u} (2.59123 \dots) = 0

\frac{d}{d u} (2.59123 \dots)

Derivative of a constant:

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

= 0

= - 16 u^{3} + 25.89299 \dots u^{2} - 3.98507 \dots u - 5.69314 \dots + 0

Simplify

= - 16 u^{3} + 25.89299 \dots u^{2} - 3.98507 \dots u - 5.69314 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.45514 \dots : Δ u_{1} = 0.45514 \dots

f (u_{0}) = - 4 \cdot 0^{4} + 8.63099 \dots \cdot 0^{3} - 1.99253 \dots \cdot 0^{2} - 5.69314 \dots \cdot 0 + 2.59123 \dots = 2.59123 \dots

f^{^{'}} (u_{0}) = - 16 \cdot 0^{3} + 25.89299 \dots \cdot 0^{2} - 3.98507 \dots \cdot 0 - 5.69314 \dots = - 5.69314 \dots

u_{1} = 0.45514 \dots

Δ u_{1} = ∣ 0.45514 \dots - 0 ∣ = 0.45514 \dots

Δ u_{1} = 0.45514 \dots

u_{2} = 0.51796 \dots : Δ u_{2} = 0.06281 \dots

f (u_{1}) = - 4 \cdot 0.45514 \dots^{4} + 8.63099 \dots \cdot 0.45514 \dots^{3} - 1.99253 \dots \cdot 0.45514 \dots^{2} - 5.69314 \dots \cdot 0.45514 \dots + 2.59123 \dots = 0.22937 \dots

f^{^{'}} (u_{1}) = - 16 \cdot 0.45514 \dots^{3} + 25.89299 \dots \cdot 0.45514 \dots^{2} - 3.98507 \dots \cdot 0.45514 \dots - 5.69314 \dots = - 3.65154 \dots

u_{2} = 0.51796 \dots

Δ u_{2} = ∣ 0.51796 \dots - 0.45514 \dots ∣ = 0.06281 \dots

Δ u_{2} = 0.06281 \dots

u_{3} = 0.52432 \dots : Δ u_{3} = 0.00635 \dots

f (u_{2}) = - 4 \cdot 0.51796 \dots^{4} + 8.63099 \dots \cdot 0.51796 \dots^{3} - 1.99253 \dots \cdot 0.51796 \dots^{2} - 5.69314 \dots \cdot 0.51796 \dots + 2.59123 \dots = 0.01929 \dots

f^{^{'}} (u_{2}) = - 16 \cdot 0.51796 \dots^{3} + 25.89299 \dots \cdot 0.51796 \dots^{2} - 3.98507 \dots \cdot 0.51796 \dots - 5.69314 \dots = - 3.03391 \dots

u_{3} = 0.52432 \dots

Δ u_{3} = ∣ 0.52432 \dots - 0.51796 \dots ∣ = 0.00635 \dots

Δ u_{3} = 0.00635 \dots

u_{4} = 0.52439 \dots : Δ u_{4} = 0.00006 \dots

f (u_{3}) = - 4 \cdot 0.52432 \dots^{4} + 8.63099 \dots \cdot 0.52432 \dots^{3} - 1.99253 \dots \cdot 0.52432 \dots^{2} - 5.69314 \dots \cdot 0.52432 \dots + 2.59123 \dots = 0.00020 \dots

f^{^{'}} (u_{3}) = - 16 \cdot 0.52432 \dots^{3} + 25.89299 \dots \cdot 0.52432 \dots^{2} - 3.98507 \dots \cdot 0.52432 \dots - 5.69314 \dots = - 2.97053 \dots

u_{4} = 0.52439 \dots

Δ u_{4} = ∣ 0.52439 \dots - 0.52432 \dots ∣ = 0.00006 \dots

Δ u_{4} = 0.00006 \dots

u_{5} = 0.52439 \dots : Δ u_{5} = 7.72366 E - 9

f (u_{4}) = - 4 \cdot 0.52439 \dots^{4} + 8.63099 \dots \cdot 0.52439 \dots^{3} - 1.99253 \dots \cdot 0.52439 \dots^{2} - 5.69314 \dots \cdot 0.52439 \dots + 2.59123 \dots = 2.29382 E - 8

f^{^{'}} (u_{4}) = - 16 \cdot 0.52439 \dots^{3} + 25.89299 \dots \cdot 0.52439 \dots^{2} - 3.98507 \dots \cdot 0.52439 \dots - 5.69314 \dots = - 2.96985 \dots

u_{5} = 0.52439 \dots

Δ u_{5} = ∣ 0.52439 \dots - 0.52439 \dots ∣ = 7.72366 E - 9

Δ u_{5} = 7.72366 E - 9

u \approx 0.52439 \dots

Apply long division:

\frac{- 4 u ^{4} + 8.63099 \dots u ^{3} - 1.99253 \dots u ^{2} - 5.69314 \dots u + 2.59123 \dots}{u - 0.52439 \dots} = - 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots

- 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots \approx 0

Find one solution for

- 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots = 0

using Newton-Raphson:

u \approx - 0.79147 \dots

- 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots

Find

f^{^{'}} (u) : - 12 u^{2} + 13.06685 \dots u + 1.43354 \dots

\frac{d}{d u} (- 4 u^{3} + 6.53342 \dots u^{2} + 1.43354 \dots u - 4.94140 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (4 u^{3}) + \frac{d}{d u} (6.53342 \dots u^{2}) + \frac{d}{d u} (1.43354 \dots u) - \frac{d}{d u} (4.94140 \dots)

\frac{d}{d u} (4 u^{3}) = 12 u^{2}

\frac{d}{d u} (4 u^{3})

Take the constant out:

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

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

Apply the Power Rule:

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

= 4 \cdot 3 u^{3 - 1}

Simplify

= 12 u^{2}

\frac{d}{d u} (6.53342 \dots u^{2}) = 13.06685 \dots u

\frac{d}{d u} (6.53342 \dots u^{2})

Take the constant out:

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

= 6.53342 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 6.53342 \dots \cdot 2 u^{2 - 1}

Simplify

= 13.06685 \dots u

\frac{d}{d u} (1.43354 \dots u) = 1.43354 \dots

\frac{d}{d u} (1.43354 \dots u)

Take the constant out:

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

= 1.43354 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.43354 \dots \cdot 1

Simplify

= 1.43354 \dots

\frac{d}{d u} (4.94140 \dots) = 0

\frac{d}{d u} (4.94140 \dots)

Derivative of a constant:

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

= 0

= - 12 u^{2} + 13.06685 \dots u + 1.43354 \dots - 0

Simplify

= - 12 u^{2} + 13.06685 \dots u + 1.43354 \dots

Let

u_{0} = 3

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 2.26016 \dots : Δ u_{1} = 0.73983 \dots

f (u_{0}) = - 4 \cdot 3^{3} + 6.53342 \dots \cdot 3^{2} + 1.43354 \dots \cdot 3 - 4.94140 \dots = - 49.83990 \dots

f^{^{'}} (u_{0}) = - 12 \cdot 3^{2} + 13.06685 \dots \cdot 3 + 1.43354 \dots = - 67.36588 \dots

u_{1} = 2.26016 \dots

Δ u_{1} = ∣ 2.26016 \dots - 3 ∣ = 0.73983 \dots

Δ u_{1} = 0.73983 \dots

u_{2} = 1.78183 \dots : Δ u_{2} = 0.47832 \dots

f (u_{1}) = - 4 \cdot 2.26016 \dots^{3} + 6.53342 \dots \cdot 2.26016 \dots^{2} + 1.43354 \dots \cdot 2.26016 \dots - 4.94140 \dots = - 14.50903 \dots

f^{^{'}} (u_{1}) = - 12 \cdot 2.26016 \dots^{2} + 13.06685 \dots \cdot 2.26016 \dots + 1.43354 \dots = - 30.33318 \dots

u_{2} = 1.78183 \dots

Δ u_{2} = ∣ 1.78183 \dots - 2.26016 \dots ∣ = 0.47832 \dots

Δ u_{2} = 0.47832 \dots

u_{3} = 1.46256 \dots : Δ u_{3} = 0.31927 \dots

f (u_{2}) = - 4 \cdot 1.78183 \dots^{3} + 6.53342 \dots \cdot 1.78183 \dots^{2} + 1.43354 \dots \cdot 1.78183 \dots - 4.94140 \dots = - 4.27274 \dots

f^{^{'}} (u_{2}) = - 12 \cdot 1.78183 \dots^{2} + 13.06685 \dots \cdot 1.78183 \dots + 1.43354 \dots = - 13.38281 \dots

u_{3} = 1.46256 \dots

Δ u_{3} = ∣ 1.46256 \dots - 1.78183 \dots ∣ = 0.31927 \dots

Δ u_{3} = 0.31927 \dots

u_{4} = 1.19261 \dots : Δ u_{4} = 0.26995 \dots

f (u_{3}) = - 4 \cdot 1.46256 \dots^{3} + 6.53342 \dots \cdot 1.46256 \dots^{2} + 1.43354 \dots \cdot 1.46256 \dots - 4.94140 \dots = - 1.38340 \dots

f^{^{'}} (u_{3}) = - 12 \cdot 1.46256 \dots^{2} + 13.06685 \dots \cdot 1.46256 \dots + 1.43354 \dots = - 5.12454 \dots

u_{4} = 1.19261 \dots

Δ u_{4} = ∣ 1.19261 \dots - 1.46256 \dots ∣ = 0.26995 \dots

Δ u_{4} = 0.26995 \dots

u_{5} = - 13.10640 \dots : Δ u_{5} = 14.29901 \dots

f (u_{4}) = - 4 \cdot 1.19261 \dots^{3} + 6.53342 \dots \cdot 1.19261 \dots^{2} + 1.43354 \dots \cdot 1.19261 \dots - 4.94140 \dots = - 0.72421 \dots

f^{^{'}} (u_{4}) = - 12 \cdot 1.19261 \dots^{2} + 13.06685 \dots \cdot 1.19261 \dots + 1.43354 \dots = - 0.05064 \dots

u_{5} = - 13.10640 \dots

Δ u_{5} = ∣ - 13.10640 \dots - 1.19261 \dots ∣ = 14.29901 \dots

Δ u_{5} = 14.29901 \dots

u_{6} = - 8.57776 \dots : Δ u_{6} = 4.52864 \dots

f (u_{5}) = - 4 (- 13.10640 \dots)^{3} + 6.53342 \dots (- 13.10640 \dots)^{2} + 1.43354 \dots (- 13.10640 \dots) - 4.94140 \dots = 10104.13392 \dots

f^{^{'}} (u_{5}) = - 12 (- 13.10640 \dots)^{2} + 13.06685 \dots (- 13.10640 \dots) + 1.43354 \dots = - 2231.16096 \dots

u_{6} = - 8.57776 \dots

Δ u_{6} = ∣ - 8.57776 \dots - (- 13.10640 \dots) ∣ = 4.52864 \dots

Δ u_{6} = 4.52864 \dots

u_{7} = - 5.57045 \dots : Δ u_{7} = 3.00730 \dots

f (u_{6}) = - 4 (- 8.57776 \dots)^{3} + 6.53342 \dots (- 8.57776 \dots)^{2} + 1.43354 \dots (- 8.57776 \dots) - 4.94140 \dots = 2988.01817 \dots

f^{^{'}} (u_{6}) = - 12 (- 8.57776 \dots)^{2} + 13.06685 \dots (- 8.57776 \dots) + 1.43354 \dots = - 993.58713 \dots

u_{7} = - 5.57045 \dots

Δ u_{7} = ∣ - 5.57045 \dots - (- 8.57776 \dots) ∣ = 3.00730 \dots

Δ u_{7} = 3.00730 \dots

u_{8} = - 3.58447 \dots : Δ u_{8} = 1.98598 \dots

f (u_{7}) = - 4 (- 5.57045 \dots)^{3} + 6.53342 \dots (- 5.57045 \dots)^{2} + 1.43354 \dots (- 5.57045 \dots) - 4.94140 \dots = 881.21142 \dots

f^{^{'}} (u_{7}) = - 12 (- 5.57045 \dots)^{2} + 13.06685 \dots (- 5.57045 \dots) + 1.43354 \dots = - 443.71510 \dots

u_{8} = - 3.58447 \dots

Δ u_{8} = ∣ - 3.58447 \dots - (- 5.57045 \dots) ∣ = 1.98598 \dots

Δ u_{8} = 1.98598 \dots

u_{9} = - 2.29137 \dots : Δ u_{9} = 1.29310 \dots

f (u_{8}) = - 4 (- 3.58447 \dots)^{3} + 6.53342 \dots (- 3.58447 \dots)^{2} + 1.43354 \dots (- 3.58447 \dots) - 4.94140 \dots = 258.08452 \dots

f^{^{'}} (u_{8}) = - 12 (- 3.58447 \dots)^{2} + 13.06685 \dots (- 3.58447 \dots) + 1.43354 \dots = - 199.58579 \dots

u_{9} = - 2.29137 \dots

Δ u_{9} = ∣ - 2.29137 \dots - (- 3.58447 \dots) ∣ = 1.29310 \dots

Δ u_{9} = 1.29310 \dots

u_{10} = - 1.48056 \dots : Δ u_{10} = 0.81081 \dots

f (u_{9}) = - 4 (- 2.29137 \dots)^{3} + 6.53342 \dots (- 2.29137 \dots)^{2} + 1.43354 \dots (- 2.29137 \dots) - 4.94140 \dots = 74.19938 \dots

f^{^{'}} (u_{9}) = - 12 (- 2.29137 \dots)^{2} + 13.06685 \dots (- 2.29137 \dots) + 1.43354 \dots = - 91.51226 \dots

u_{10} = - 1.48056 \dots

Δ u_{10} = ∣ - 1.48056 \dots - (- 2.29137 \dots) ∣ = 0.81081 \dots

Δ u_{10} = 0.81081 \dots

u_{11} = - 1.02282 \dots : Δ u_{11} = 0.45773 \dots

f (u_{10}) = - 4 (- 1.48056 \dots)^{3} + 6.53342 \dots (- 1.48056 \dots)^{2} + 1.43354 \dots (- 1.48056 \dots) - 4.94140 \dots = 20.23972 \dots

f^{^{'}} (u_{10}) = - 12 (- 1.48056 \dots)^{2} + 13.06685 \dots (- 1.48056 \dots) + 1.43354 \dots = - 44.21745 \dots

u_{11} = - 1.02282 \dots

Δ u_{11} = ∣ - 1.02282 \dots - (- 1.48056 \dots) ∣ = 0.45773 \dots

Δ u_{11} = 0.45773 \dots

u_{12} = - 0.83056 \dots : Δ u_{12} = 0.19226 \dots

f (u_{11}) = - 4 (- 1.02282 \dots)^{3} + 6.53342 \dots (- 1.02282 \dots)^{2} + 1.43354 \dots (- 1.02282 \dots) - 4.94140 \dots = 4.70771 \dots

f^{^{'}} (u_{11}) = - 12 (- 1.02282 \dots)^{2} + 13.06685 \dots (- 1.02282 \dots) + 1.43354 \dots = - 24.48576 \dots

u_{12} = - 0.83056 \dots

Δ u_{12} = ∣ - 0.83056 \dots - (- 1.02282 \dots) ∣ = 0.19226 \dots

Δ u_{12} = 0.19226 \dots

u_{13} = - 0.79288 \dots : Δ u_{13} = 0.03767 \dots

f (u_{12}) = - 4 (- 0.83056 \dots)^{3} + 6.53342 \dots (- 0.83056 \dots)^{2} + 1.43354 \dots (- 0.83056 \dots) - 4.94140 \dots = 0.66678 \dots

f^{^{'}} (u_{12}) = - 12 (- 0.83056 \dots)^{2} + 13.06685 \dots (- 0.83056 \dots) + 1.43354 \dots = - 17.69741 \dots

u_{13} = - 0.79288 \dots

Δ u_{13} = ∣ - 0.79288 \dots - (- 0.83056 \dots) ∣ = 0.03767 \dots

Δ u_{13} = 0.03767 \dots

u_{14} = - 0.79147 \dots : Δ u_{14} = 0.00140 \dots

f (u_{13}) = - 4 (- 0.79288 \dots)^{3} + 6.53342 \dots (- 0.79288 \dots)^{2} + 1.43354 \dots (- 0.79288 \dots) - 4.94140 \dots = 0.0232093455

f^{^{'}} (u_{13}) = - 12 (- 0.79288 \dots)^{2} + 13.06685 \dots (- 0.79288 \dots) + 1.43354 \dots = - 16.47108 \dots

u_{14} = - 0.79147 \dots

Δ u_{14} = ∣ - 0.79147 \dots - (- 0.79288 \dots) ∣ = 0.00140 \dots

Δ u_{14} = 0.00140 \dots

u_{15} = - 0.79147 \dots : Δ u_{15} = 1.9392 E - 6

f (u_{14}) = - 4 (- 0.79147 \dots)^{3} + 6.53342 \dots (- 0.79147 \dots)^{2} + 1.43354 \dots (- 0.79147 \dots) - 4.94140 \dots = 0.00003 \dots

f^{^{'}} (u_{14}) = - 12 (- 0.79147 \dots)^{2} + 13.06685 \dots (- 0.79147 \dots) + 1.43354 \dots = - 16.42588 \dots

u_{15} = - 0.79147 \dots

Δ u_{15} = ∣ - 0.79147 \dots - (- 0.79147 \dots) ∣ = 1.9392 E - 6

Δ u_{15} = 1.9392 E - 6

u_{16} = - 0.79147 \dots : Δ u_{16} = 3.6702 E - 12

f (u_{15}) = - 4 (- 0.79147 \dots)^{3} + 6.53342 \dots (- 0.79147 \dots)^{2} + 1.43354 \dots (- 0.79147 \dots) - 4.94140 \dots = 6.0286 E - 11

f^{^{'}} (u_{15}) = - 12 (- 0.79147 \dots)^{2} + 13.06685 \dots (- 0.79147 \dots) + 1.43354 \dots = - 16.42581 \dots

u_{16} = - 0.79147 \dots

Δ u_{16} = ∣ - 0.79147 \dots - (- 0.79147 \dots) ∣ = 3.6702 E - 12

Δ u_{16} = 3.6702 E - 12

u \approx - 0.79147 \dots

Apply long division:

\frac{- 4 u ^{3} + 6.53342 \dots u ^{2} + 1.43354 \dots u - 4.94140 \dots}{u + 0.79147 \dots} = - 4 u^{2} + 9.69933 \dots u - 6.24326 \dots

- 4 u^{2} + 9.69933 \dots u - 6.24326 \dots \approx 0

Find one solution for

- 4 u^{2} + 9.69933 \dots u - 6.24326 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

- 4 u^{2} + 9.69933 \dots u - 6.24326 \dots = 0

Newton-Raphson Approximation Definition

f (u) = - 4 u^{2} + 9.69933 \dots u - 6.24326 \dots

Find

f^{^{'}} (u) : - 8 u + 9.69933 \dots

\frac{d}{d u} (- 4 u^{2} + 9.69933 \dots u - 6.24326 \dots)

Apply the Sum/Difference Rule:

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

= - \frac{d}{d u} (4 u^{2}) + \frac{d}{d u} (9.69933 \dots u) - \frac{d}{d u} (6.24326 \dots)

\frac{d}{d u} (4 u^{2}) = 8 u

\frac{d}{d u} (4 u^{2})

Take the constant out:

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

= 4 \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 4 \cdot 2 u^{2 - 1}

Simplify

= 8 u

\frac{d}{d u} (9.69933 \dots u) = 9.69933 \dots

\frac{d}{d u} (9.69933 \dots u)

Take the constant out:

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

= 9.69933 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 9.69933 \dots \cdot 1

Simplify

= 9.69933 \dots

\frac{d}{d u} (6.24326 \dots) = 0

\frac{d}{d u} (6.24326 \dots)

Derivative of a constant:

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

= 0

= - 8 u + 9.69933 \dots - 0

Simplify

= - 8 u + 9.69933 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 1.32008 \dots : Δ u_{1} = 0.32008 \dots

f (u_{0}) = - 4 \cdot 1^{2} + 9.69933 \dots \cdot 1 - 6.24326 \dots = - 0.54392 \dots

f^{^{'}} (u_{0}) = - 8 \cdot 1 + 9.69933 \dots = 1.69933 \dots

u_{1} = 1.32008 \dots

Δ u_{1} = ∣ 1.32008 \dots - 1 ∣ = 0.32008 \dots

Δ u_{1} = 0.32008 \dots

u_{2} = 0.84428 \dots : Δ u_{2} = 0.47579 \dots

f (u_{1}) = - 4 \cdot 1.32008 \dots^{2} + 9.69933 \dots \cdot 1.32008 \dots - 6.24326 \dots = - 0.40980 \dots

f^{^{'}} (u_{1}) = - 8 \cdot 1.32008 \dots + 9.69933 \dots = - 0.86130 \dots

u_{2} = 0.84428 \dots

Δ u_{2} = ∣ 0.84428 \dots - 1.32008 \dots ∣ = 0.47579 \dots

Δ u_{2} = 0.47579 \dots

u_{3} = 1.15175 \dots : Δ u_{3} = 0.30747 \dots

f (u_{2}) = - 4 \cdot 0.84428 \dots^{2} + 9.69933 \dots \cdot 0.84428 \dots - 6.24326 \dots = - 0.90553 \dots

f^{^{'}} (u_{2}) = - 8 \cdot 0.84428 \dots + 9.69933 \dots = 2.94507 \dots

u_{3} = 1.15175 \dots

Δ u_{3} = ∣ 1.15175 \dots - 0.84428 \dots ∣ = 0.30747 \dots

Δ u_{3} = 0.30747 \dots

u_{4} = 1.93099 \dots : Δ u_{4} = 0.77924 \dots

f (u_{3}) = - 4 \cdot 1.15175 \dots^{2} + 9.69933 \dots \cdot 1.15175 \dots - 6.24326 \dots = - 0.37815 \dots

f^{^{'}} (u_{3}) = - 8 \cdot 1.15175 \dots + 9.69933 \dots = 0.48529 \dots

u_{4} = 1.93099 \dots

Δ u_{4} = ∣ 1.93099 \dots - 1.15175 \dots ∣ = 0.77924 \dots

Δ u_{4} = 0.77924 \dots

u_{5} = 1.50848 \dots : Δ u_{5} = 0.42251 \dots

f (u_{4}) = - 4 \cdot 1.93099 \dots^{2} + 9.69933 \dots \cdot 1.93099 \dots - 6.24326 \dots = - 2.42887 \dots

f^{^{'}} (u_{4}) = - 8 \cdot 1.93099 \dots + 9.69933 \dots = - 5.74865 \dots

u_{5} = 1.50848 \dots

Δ u_{5} = ∣ 1.50848 \dots - 1.93099 \dots ∣ = 0.42251 \dots

Δ u_{5} = 0.42251 \dots

u_{6} = 1.20700 \dots : Δ u_{6} = 0.30147 \dots

f (u_{5}) = - 4 \cdot 1.50848 \dots^{2} + 9.69933 \dots \cdot 1.50848 \dots - 6.24326 \dots = - 0.71406 \dots

f^{^{'}} (u_{5}) = - 8 \cdot 1.50848 \dots + 9.69933 \dots = - 2.36855 \dots

u_{6} = 1.20700 \dots

Δ u_{6} = ∣ 1.20700 \dots - 1.50848 \dots ∣ = 0.30147 \dots

Δ u_{6} = 0.30147 \dots

u_{7} = 9.60809 \dots : Δ u_{7} = 8.40108 \dots

f (u_{6}) = - 4 \cdot 1.20700 \dots^{2} + 9.69933 \dots \cdot 1.20700 \dots - 6.24326 \dots = - 0.36355 \dots

f^{^{'}} (u_{6}) = - 8 \cdot 1.20700 \dots + 9.69933 \dots = 0.04327 \dots

u_{7} = 9.60809 \dots

Δ u_{7} = ∣ 9.60809 \dots - 1.20700 \dots ∣ = 8.40108 \dots

Δ u_{7} = 8.40108 \dots

u_{8} = 5.40484 \dots : Δ u_{8} = 4.20324 \dots

f (u_{7}) = - 4 \cdot 9.60809 \dots^{2} + 9.69933 \dots \cdot 9.60809 \dots - 6.24326 \dots = - 282.31282 \dots

f^{^{'}} (u_{7}) = - 8 \cdot 9.60809 \dots + 9.69933 \dots = - 67.16539 \dots

u_{8} = 5.40484 \dots

Δ u_{8} = ∣ 5.40484 \dots - 9.60809 \dots ∣ = 4.20324 \dots

Δ u_{8} = 4.20324 \dots

u_{9} = 3.29779 \dots : Δ u_{9} = 2.10704 \dots

f (u_{8}) = - 4 \cdot 5.40484 \dots^{2} + 9.69933 \dots \cdot 5.40484 \dots - 6.24326 \dots = - 70.66918 \dots

f^{^{'}} (u_{8}) = - 8 \cdot 5.40484 \dots + 9.69933 \dots = - 33.53940 \dots

u_{9} = 3.29779 \dots

Δ u_{9} = ∣ 3.29779 \dots - 5.40484 \dots ∣ = 2.10704 \dots

Δ u_{9} = 2.10704 \dots

u_{10} = 2.23332 \dots : Δ u_{10} = 1.06447 \dots

f (u_{9}) = - 4 \cdot 3.29779 \dots^{2} + 9.69933 \dots \cdot 3.29779 \dots - 6.24326 \dots = - 17.75862 \dots

f^{^{'}} (u_{9}) = - 8 \cdot 3.29779 \dots + 9.69933 \dots = - 16.68301 \dots

u_{10} = 2.23332 \dots

Δ u_{10} = ∣ 2.23332 \dots - 3.29779 \dots ∣ = 1.06447 \dots

Δ u_{10} = 1.06447 \dots

u_{11} = 1.67836 \dots : Δ u_{11} = 0.55495 \dots

f (u_{10}) = - 4 \cdot 2.23332 \dots^{2} + 9.69933 \dots \cdot 2.23332 \dots - 6.24326 \dots = - 4.53241 \dots

f^{^{'}} (u_{10}) = - 8 \cdot 2.23332 \dots + 9.69933 \dots = - 8.16722 \dots

u_{11} = 1.67836 \dots

Δ u_{11} = ∣ 1.67836 \dots - 2.23332 \dots ∣ = 0.55495 \dots

Δ u_{11} = 0.55495 \dots

u_{12} = 1.34789 \dots : Δ u_{12} = 0.33047 \dots

f (u_{11}) = - 4 \cdot 1.67836 \dots^{2} + 9.69933 \dots \cdot 1.67836 \dots - 6.24326 \dots = - 1.23188 \dots

f^{^{'}} (u_{11}) = - 8 \cdot 1.67836 \dots + 9.69933 \dots = - 3.72761 \dots

u_{12} = 1.34789 \dots

Δ u_{12} = ∣ 1.34789 \dots - 1.67836 \dots ∣ = 0.33047 \dots

Δ u_{12} = 0.33047 \dots

u_{13} = 0.94482 \dots : Δ u_{13} = 0.40307 \dots

f (u_{12}) = - 4 \cdot 1.34789 \dots^{2} + 9.69933 \dots \cdot 1.34789 \dots - 6.24326 \dots = - 0.43685 \dots

f^{^{'}} (u_{12}) = - 8 \cdot 1.34789 \dots + 9.69933 \dots = - 1.08381 \dots

u_{13} = 0.94482 \dots

Δ u_{13} = ∣ 0.94482 \dots - 1.34789 \dots ∣ = 0.40307 \dots

Δ u_{13} = 0.40307 \dots

Cannot find solution

The solutions are

u \approx - 1.15774 \dots, u \approx 0.52439 \dots, u \approx - 0.79147 \dots

u \approx - 1.15774 \dots, u \approx 0.52439 \dots, u \approx - 0.79147 \dots

Verify Solutions:

u \approx - 1.15774 \dots

False

, u \approx 0.52439 \dots

True

, u \approx - 0.79147 \dots

True

Check the solutions by plugging them into

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

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

Plug in

u \approx - 1.15774 \dots :

False

- 1 + (1 - (- 1.15774 \dots)^{2}) \cdot 2 1 - (- 1.15774 \dots) = 0

- 1 + (1 - (- 1.15774 \dots)^{2}) \cdot 2 1 - (- 1.15774 \dots) = - 2

- 1 + (1 - (- 1.15774 \dots)^{2}) \cdot 2 1 - (- 1.15774 \dots)

Apply rule

- (- a) = a

= - 1 + (1 - (- 1.15774 \dots)^{2}) \cdot 2 1 + 1.15774 \dots

(1 - (- 1.15774 \dots)^{2}) \cdot 2 1 + 1.15774 \dots = - 0.68076 \dots 2.15774 \dots

(1 - (- 1.15774 \dots)^{2}) \cdot 2 1 + 1.15774 \dots

(- 1.15774 \dots)^{2} = 1.34038 \dots

(- 1.15774 \dots)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 1.15774 \dots)^{2} = 1.15774 \dots^{2}

= 1.15774 \dots^{2}

1.15774 \dots^{2} = 1.34038 \dots

= 1.34038 \dots

= 2 (1 - 1.34038 \dots) 1 + 1.15774 \dots

Add the numbers:

1 + 1.15774 \dots = 2.15774 \dots

= 2 2.15774 \dots (1 - 1.34038 \dots)

Subtract the numbers:

1 - 1.34038 \dots = - 0.34038 \dots

= 2 (- 0.34038 \dots) 2.15774 \dots

Remove parentheses:

(- a) = - a

= - 0.34038 \dots \cdot 2 2.15774 \dots

Multiply the numbers:

0.34038 \dots \cdot 2 = 0.68076 \dots

= - 0.68076 \dots 2.15774 \dots

= - 1 - 0.68076 \dots 2.15774 \dots

0.68076 \dots 2.15774 \dots = 1

0.68076 \dots 2.15774 \dots

2.15774 \dots = 1.46892 \dots

= 0.68076 \dots \cdot 1.46892 \dots

Multiply the numbers:

0.68076 \dots \cdot 1.46892 \dots = 1

= 1

= - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2

- 2 = 0

False

Plug in

u \approx 0.52439 \dots :

True

- 1 + (1 - 0.52439 \dots^{2}) \cdot 2 1 - 0.52439 \dots = 0

- 1 + (1 - 0.52439 \dots^{2}) \cdot 2 1 - 0.52439 \dots = 5.0 E - 15

- 1 + (1 - 0.52439 \dots^{2}) \cdot 2 1 - 0.52439 \dots

(1 - 0.52439 \dots^{2}) \cdot 2 1 - 0.52439 \dots = 1.45002 \dots 0.47560 \dots

(1 - 0.52439 \dots^{2}) \cdot 2 1 - 0.52439 \dots

0.52439 \dots^{2} = 0.27498 \dots

= 2 (1 - 0.27498 \dots) 1 - 0.52439 \dots

Subtract the numbers:

1 - 0.52439 \dots = 0.47560 \dots

= 2 0.47560 \dots (1 - 0.27498 \dots)

Subtract the numbers:

1 - 0.27498 \dots = 0.72501 \dots

= 2 \cdot 0.72501 \dots 0.47560 \dots

Multiply the numbers:

0.72501 \dots \cdot 2 = 1.45002 \dots

= 1.45002 \dots 0.47560 \dots

= - 1 + 1.45002 \dots 0.47560 \dots

1.45002 \dots 0.47560 \dots = 1

1.45002 \dots 0.47560 \dots

0.47560 \dots = 0.68964 \dots

= 0.68964 \dots \cdot 1.45002 \dots

Multiply the numbers:

1.45002 \dots \cdot 0.68964 \dots = 1

= 1

= - 1 + 1

Add/Subtract the numbers:

- 1 + 1 = 5.0 E - 15

= 5.0 E - 15

5.0 E - 15 = 0

True

Plug in

u \approx - 0.79147 \dots :

True

- 1 + (1 - (- 0.79147 \dots)^{2}) \cdot 2 1 - (- 0.79147 \dots) = 0

- 1 + (1 - (- 0.79147 \dots)^{2}) \cdot 2 1 - (- 0.79147 \dots) = 5.0 E - 15

- 1 + (1 - (- 0.79147 \dots)^{2}) \cdot 2 1 - (- 0.79147 \dots)

Apply rule

- (- a) = a

= - 1 + (1 - (- 0.79147 \dots)^{2}) \cdot 2 1 + 0.79147 \dots

(1 - (- 0.79147 \dots)^{2}) \cdot 2 1 + 0.79147 \dots = 0.74712 \dots 1.79147 \dots

(1 - (- 0.79147 \dots)^{2}) \cdot 2 1 + 0.79147 \dots

(- 0.79147 \dots)^{2} = 0.62643 \dots

(- 0.79147 \dots)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 0.79147 \dots)^{2} = 0.79147 \dots^{2}

= 0.79147 \dots^{2}

0.79147 \dots^{2} = 0.62643 \dots

= 0.62643 \dots

= 2 (1 - 0.62643 \dots) 1 + 0.79147 \dots

Add the numbers:

1 + 0.79147 \dots = 1.79147 \dots

= 2 1.79147 \dots (1 - 0.62643 \dots)

Subtract the numbers:

1 - 0.62643 \dots = 0.37356 \dots

= 2 \cdot 0.37356 \dots 1.79147 \dots

Multiply the numbers:

0.37356 \dots \cdot 2 = 0.74712 \dots

= 0.74712 \dots 1.79147 \dots

= - 1 + 0.74712 \dots 1.79147 \dots

0.74712 \dots 1.79147 \dots = 1

0.74712 \dots 1.79147 \dots

1.79147 \dots = 1.33846 \dots

= 0.74712 \dots \cdot 1.33846 \dots

Multiply the numbers:

0.74712 \dots \cdot 1.33846 \dots = 1

= 1

= - 1 + 1

Add/Subtract the numbers:

- 1 + 1 = 5.0 E - 15

= 5.0 E - 15

5.0 E - 15 = 0

True

The solutions are

u \approx 0.52439 \dots, u \approx - 0.79147 \dots

Substitute back

u = cos (x)

cos (x) \approx 0.52439 \dots, cos (x) \approx - 0.79147 \dots

cos (x) \approx 0.52439 \dots, cos (x) \approx - 0.79147 \dots

cos (x) = 0.52439 \dots : x = arccos (0.52439 \dots) + 2 πn, x = 2 π - arccos (0.52439 \dots) + 2 πn

cos (x) = 0.52439 \dots

Apply trig inverse properties

cos (x) = 0.52439 \dots

General solutions for

cos (x) = 0.52439 \dots

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

x = arccos (0.52439 \dots) + 2 πn, x = 2 π - arccos (0.52439 \dots) + 2 πn

x = arccos (0.52439 \dots) + 2 πn, x = 2 π - arccos (0.52439 \dots) + 2 πn

cos (x) = - 0.79147 \dots : x = arccos (- 0.79147 \dots) + 2 πn, x = - arccos (- 0.79147 \dots) + 2 πn

cos (x) = - 0.79147 \dots

Apply trig inverse properties

cos (x) = - 0.79147 \dots

General solutions for

cos (x) = - 0.79147 \dots

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

x = arccos (- 0.79147 \dots) + 2 πn, x = - arccos (- 0.79147 \dots) + 2 πn

x = arccos (- 0.79147 \dots) + 2 πn, x = - arccos (- 0.79147 \dots) + 2 πn

Combine all the solutions

x = arccos (0.52439 \dots) + 2 πn, x = 2 π - arccos (0.52439 \dots) + 2 πn, x = arccos (- 0.79147 \dots) + 2 πn, x = - arccos (- 0.79147 \dots) + 2 πn

Show solutions in decimal form

x = 1.01879 \dots + 2 πn, x = 2 π - 1.01879 \dots + 2 πn, x = 2.48401 \dots + 2 πn, x = - 2.48401 \dots + 2 πn

Graph

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