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2cos(x)cos^3(x)+cos^2(x)-sin^2(x)=cos(x)

Solution

2 cos (x) cos^{3} (x) + cos^{2} (x) - sin^{2} (x) = cos (x)

Solution

x = 2.05721 \dots + 2 πn, x = - 2.05721 \dots + 2 πn, x = 0.72493 \dots + 2 πn, x = 2 π - 0.72493 \dots + 2 πn

Degrees

x = 117.86956 \dots^{\circ} + 36 0^{\circ} n, x = - 117.86956 \dots^{\circ} + 36 0^{\circ} n, x = 41.53566 \dots^{\circ} + 36 0^{\circ} n, x = 318.46433 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos (x) cos^{3} (x) + cos^{2} (x) - sin^{2} (x) = cos (x)

Subtract

cos (x)

from both sides

2 cos^{4} (x) + cos^{2} (x) - sin^{2} (x) - cos (x) = 0

Rewrite using trig identities

- cos (x) + cos^{2} (x) - sin^{2} (x) + 2 cos^{4} (x)

Use the Pythagorean identity:

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

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

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

Simplify

- cos (x) + cos^{2} (x) - (1 - cos^{2} (x)) + 2 cos^{4} (x) : - cos (x) + 2 cos^{2} (x) + 2 cos^{4} (x) - 1

- cos (x) + cos^{2} (x) - (1 - cos^{2} (x)) + 2 cos^{4} (x)

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

Simplify

- cos (x) + cos^{2} (x) - 1 + cos^{2} (x) + 2 cos^{4} (x) : - cos (x) + 2 cos^{2} (x) + 2 cos^{4} (x) - 1

- cos (x) + cos^{2} (x) - 1 + cos^{2} (x) + 2 cos^{4} (x)

Group like terms

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

Add similar elements:

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

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

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

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

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

Solve by substitution

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

Let:

cos (x) = u

- 1 - u + 2 u^{2} + 2 u^{4} = 0

- 1 - u + 2 u^{2} + 2 u^{4} = 0 : u \approx - 0.46746 \dots, u \approx 0.74854 \dots

- 1 - u + 2 u^{2} + 2 u^{4} = 0

Write in the standard form

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

2 u^{4} + 2 u^{2} - u - 1 = 0

Find one solution for

2 u^{4} + 2 u^{2} - u - 1 = 0

using Newton-Raphson:

u \approx - 0.46746 \dots

2 u^{4} + 2 u^{2} - u - 1 = 0

Newton-Raphson Approximation Definition

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

Find

f^{^{'}} (u) : 8 u^{3} + 4 u - 1

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

Apply the Sum/Difference Rule:

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

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

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

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

Take the constant out:

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

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

Apply the Power Rule:

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

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

Simplify

= 8 u^{3}

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

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

Take the constant out:

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

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

Apply the Power Rule:

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

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

Simplify

= 4 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

= 8 u^{3} + 4 u - 1 - 0

Simplify

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

Let

u_{0} = - 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 0.69230 \dots : Δ u_{1} = 0.30769 \dots

f (u_{0}) = 2 (- 1)^{4} + 2 (- 1)^{2} - (- 1) - 1 = 4

f^{^{'}} (u_{0}) = 8 (- 1)^{3} + 4 (- 1) - 1 = - 13

u_{1} = - 0.69230 \dots

Δ u_{1} = ∣ - 0.69230 \dots - (- 1) ∣ = 0.30769 \dots

Δ u_{1} = 0.30769 \dots

u_{2} = - 0.51946 \dots : Δ u_{2} = 0.17284 \dots

f (u_{1}) = 2 (- 0.69230 \dots)^{4} + 2 (- 0.69230 \dots)^{2} - (- 0.69230 \dots) - 1 = 1.11032 \dots

f^{^{'}} (u_{1}) = 8 (- 0.69230 \dots)^{3} + 4 (- 0.69230 \dots) - 1 = - 6.42375 \dots

u_{2} = - 0.51946 \dots

Δ u_{2} = ∣ - 0.51946 \dots - (- 0.69230 \dots) ∣ = 0.17284 \dots

Δ u_{2} = 0.17284 \dots

u_{3} = - 0.47069 \dots : Δ u_{3} = 0.04876 \dots

f (u_{2}) = 2 (- 0.51946 \dots)^{4} + 2 (- 0.51946 \dots)^{2} - (- 0.51946 \dots) - 1 = 0.20476 \dots

f^{^{'}} (u_{2}) = 8 (- 0.51946 \dots)^{3} + 4 (- 0.51946 \dots) - 1 = - 4.19921 \dots

u_{3} = - 0.47069 \dots

Δ u_{3} = ∣ - 0.47069 \dots - (- 0.51946 \dots) ∣ = 0.04876 \dots

Δ u_{3} = 0.04876 \dots

u_{4} = - 0.46747 \dots : Δ u_{4} = 0.00322 \dots

f (u_{3}) = 2 (- 0.47069 \dots)^{4} + 2 (- 0.47069 \dots)^{2} - (- 0.47069 \dots) - 1 = 0.01198 \dots

f^{^{'}} (u_{3}) = 8 (- 0.47069 \dots)^{3} + 4 (- 0.47069 \dots) - 1 = - 3.71707 \dots

u_{4} = - 0.46747 \dots

Δ u_{4} = ∣ - 0.46747 \dots - (- 0.47069 \dots) ∣ = 0.00322 \dots

Δ u_{4} = 0.00322 \dots

u_{5} = - 0.46746 \dots : Δ u_{5} = 0.00001 \dots

f (u_{4}) = 2 (- 0.46747 \dots)^{4} + 2 (- 0.46747 \dots)^{2} - (- 0.46747 \dots) - 1 = 0.00004 \dots

f^{^{'}} (u_{4}) = 8 (- 0.46747 \dots)^{3} + 4 (- 0.46747 \dots) - 1 = - 3.68715 \dots

u_{5} = - 0.46746 \dots

Δ u_{5} = ∣ - 0.46746 \dots - (- 0.46747 \dots) ∣ = 0.00001 \dots

Δ u_{5} = 0.00001 \dots

u_{6} = - 0.46746 \dots : Δ u_{6} = 2.15174 E - 10

f (u_{5}) = 2 (- 0.46746 \dots)^{4} + 2 (- 0.46746 \dots)^{2} - (- 0.46746 \dots) - 1 = 7.93353 E - 10

f^{^{'}} (u_{5}) = 8 (- 0.46746 \dots)^{3} + 4 (- 0.46746 \dots) - 1 = - 3.68703 \dots

u_{6} = - 0.46746 \dots

Δ u_{6} = ∣ - 0.46746 \dots - (- 0.46746 \dots) ∣ = 2.15174 E - 10

Δ u_{6} = 2.15174 E - 10

u \approx - 0.46746 \dots

Apply long division:

\frac{2 u ^{4} + 2 u ^{2} - u - 1}{u + 0.46746 \dots} = 2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots

2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots \approx 0

Find one solution for

2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots = 0

using Newton-Raphson:

u \approx 0.74854 \dots

2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots

Find

f^{^{'}} (u) : 6 u^{2} - 1.86984 \dots u + 2.43703 \dots

\frac{d}{d u} (2 u^{3} - 0.93492 \dots u^{2} + 2.43703 \dots u - 2.13921 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (2 u^{3}) - \frac{d}{d u} (0.93492 \dots u^{2}) + \frac{d}{d u} (2.43703 \dots u) - \frac{d}{d u} (2.13921 \dots)

\frac{d}{d u} (2 u^{3}) = 6 u^{2}

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

Take the constant out:

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

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

Apply the Power Rule:

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

= 2 \cdot 3 u^{3 - 1}

Simplify

= 6 u^{2}

\frac{d}{d u} (0.93492 \dots u^{2}) = 1.86984 \dots u

\frac{d}{d u} (0.93492 \dots u^{2})

Take the constant out:

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

= 0.93492 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

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

= 0.93492 \dots \cdot 2 u^{2 - 1}

Simplify

= 1.86984 \dots u

\frac{d}{d u} (2.43703 \dots u) = 2.43703 \dots

\frac{d}{d u} (2.43703 \dots u)

Take the constant out:

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

= 2.43703 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 2.43703 \dots \cdot 1

Simplify

= 2.43703 \dots

\frac{d}{d u} (2.13921 \dots) = 0

\frac{d}{d u} (2.13921 \dots)

Derivative of a constant:

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

= 0

= 6 u^{2} - 1.86984 \dots u + 2.43703 \dots - 0

Simplify

= 6 u^{2} - 1.86984 \dots u + 2.43703 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.79246 \dots : Δ u_{1} = 0.20753 \dots

f (u_{0}) = 2 \cdot 1^{3} - 0.93492 \dots \cdot 1^{2} + 2.43703 \dots \cdot 1 - 2.13921 \dots = 1.36289 \dots

f^{^{'}} (u_{0}) = 6 \cdot 1^{2} - 1.86984 \dots \cdot 1 + 2.43703 \dots = 6.56719 \dots

u_{1} = 0.79246 \dots

Δ u_{1} = ∣ 0.79246 \dots - 1 ∣ = 0.20753 \dots

Δ u_{1} = 0.20753 \dots

u_{2} = 0.75006 \dots : Δ u_{2} = 0.04240 \dots

f (u_{1}) = 2 \cdot 0.79246 \dots^{3} - 0.93492 \dots \cdot 0.79246 \dots^{2} + 2.43703 \dots \cdot 0.79246 \dots - 2.13921 \dots = 0.20027 \dots

f^{^{'}} (u_{1}) = 6 \cdot 0.79246 \dots^{2} - 1.86984 \dots \cdot 0.79246 \dots + 2.43703 \dots = 4.72328 \dots

u_{2} = 0.75006 \dots

Δ u_{2} = ∣ 0.75006 \dots - 0.79246 \dots ∣ = 0.04240 \dots

Δ u_{2} = 0.04240 \dots

u_{3} = 0.74854 \dots : Δ u_{3} = 0.00152 \dots

f (u_{2}) = 2 \cdot 0.75006 \dots^{3} - 0.93492 \dots \cdot 0.75006 \dots^{2} + 2.43703 \dots \cdot 0.75006 \dots - 2.13921 \dots = 0.00671 \dots

f^{^{'}} (u_{2}) = 6 \cdot 0.75006 \dots^{2} - 1.86984 \dots \cdot 0.75006 \dots + 2.43703 \dots = 4.41013 \dots

u_{3} = 0.74854 \dots

Δ u_{3} = ∣ 0.74854 \dots - 0.75006 \dots ∣ = 0.00152 \dots

Δ u_{3} = 0.00152 \dots

u_{4} = 0.74854 \dots : Δ u_{4} = 1.87746 E - 6

f (u_{3}) = 2 \cdot 0.74854 \dots^{3} - 0.93492 \dots \cdot 0.74854 \dots^{2} + 2.43703 \dots \cdot 0.74854 \dots - 2.13921 \dots = 8.2595 E - 6

f^{^{'}} (u_{3}) = 6 \cdot 0.74854 \dots^{2} - 1.86984 \dots \cdot 0.74854 \dots + 2.43703 \dots = 4.39929 \dots

u_{4} = 0.74854 \dots

Δ u_{4} = ∣ 0.74854 \dots - 0.74854 \dots ∣ = 1.87746 E - 6

Δ u_{4} = 1.87746 E - 6

u_{5} = 0.74854 \dots : Δ u_{5} = 2.8495 E - 12

f (u_{4}) = 2 \cdot 0.74854 \dots^{3} - 0.93492 \dots \cdot 0.74854 \dots^{2} + 2.43703 \dots \cdot 0.74854 \dots - 2.13921 \dots = 1.25358 E - 11

f^{^{'}} (u_{4}) = 6 \cdot 0.74854 \dots^{2} - 1.86984 \dots \cdot 0.74854 \dots + 2.43703 \dots = 4.39928 \dots

u_{5} = 0.74854 \dots

Δ u_{5} = ∣ 0.74854 \dots - 0.74854 \dots ∣ = 2.8495 E - 12

Δ u_{5} = 2.8495 E - 12

u \approx 0.74854 \dots

Apply long division:

\frac{2 u ^{3} - 0.93492 \dots u ^{2} + 2.43703 \dots u - 2.13921 \dots}{u - 0.74854 \dots} = 2 u^{2} + 0.56216 \dots u + 2.85784 \dots

2 u^{2} + 0.56216 \dots u + 2.85784 \dots \approx 0

Find one solution for

2 u^{2} + 0.56216 \dots u + 2.85784 \dots = 0

using Newton-Raphson:

No Solution for

u \in R

2 u^{2} + 0.56216 \dots u + 2.85784 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 2 u^{2} + 0.56216 \dots u + 2.85784 \dots

Find

f^{^{'}} (u) : 4 u + 0.56216 \dots

\frac{d}{d u} (2 u^{2} + 0.56216 \dots u + 2.85784 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (2 u^{2}) + \frac{d}{d u} (0.56216 \dots u) + \frac{d}{d u} (2.85784 \dots)

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

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

Take the constant out:

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

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

Apply the Power Rule:

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

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

Simplify

= 4 u

\frac{d}{d u} (0.56216 \dots u) = 0.56216 \dots

\frac{d}{d u} (0.56216 \dots u)

Take the constant out:

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

= 0.56216 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 0.56216 \dots \cdot 1

Simplify

= 0.56216 \dots

\frac{d}{d u} (2.85784 \dots) = 0

\frac{d}{d u} (2.85784 \dots)

Derivative of a constant:

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

= 0

= 4 u + 0.56216 \dots + 0

Simplify

= 4 u + 0.56216 \dots

Let

u_{0} = - 5

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = - 2.42527 \dots : Δ u_{1} = 2.57472 \dots

f (u_{0}) = 2 (- 5)^{2} + 0.56216 \dots (- 5) + 2.85784 \dots = 50.04701 \dots

f^{^{'}} (u_{0}) = 4 (- 5) + 0.56216 \dots = - 19.43783 \dots

u_{1} = - 2.42527 \dots

Δ u_{1} = ∣ - 2.42527 \dots - (- 5) ∣ = 2.57472 \dots

Δ u_{1} = 2.57472 \dots

u_{2} = - 0.97452 \dots : Δ u_{2} = 1.45075 \dots

f (u_{1}) = 2 (- 2.42527 \dots)^{2} + 0.56216 \dots (- 2.42527 \dots) + 2.85784 \dots = 13.25838 \dots

f^{^{'}} (u_{1}) = 4 (- 2.42527 \dots) + 0.56216 \dots = - 9.13894 \dots

u_{2} = - 0.97452 \dots

Δ u_{2} = ∣ - 0.97452 \dots - (- 2.42527 \dots) ∣ = 1.45075 \dots

Δ u_{2} = 1.45075 \dots

u_{3} = 0.28731 \dots : Δ u_{3} = 1.26183 \dots

f (u_{2}) = 2 (- 0.97452 \dots)^{2} + 0.56216 \dots (- 0.97452 \dots) + 2.85784 \dots = 4.20938 \dots

f^{^{'}} (u_{2}) = 4 (- 0.97452 \dots) + 0.56216 \dots = - 3.33592 \dots

u_{3} = 0.28731 \dots

Δ u_{3} = ∣ 0.28731 \dots - (- 0.97452 \dots) ∣ = 1.26183 \dots

Δ u_{3} = 1.26183 \dots

u_{4} = - 1.57339 \dots : Δ u_{4} = 1.86071 \dots

f (u_{3}) = 2 \cdot 0.28731 \dots^{2} + 0.56216 \dots \cdot 0.28731 \dots + 2.85784 \dots = 3.18445 \dots

f^{^{'}} (u_{3}) = 4 \cdot 0.28731 \dots + 0.56216 \dots = 1.71142 \dots

u_{4} = - 1.57339 \dots

Δ u_{4} = ∣ - 1.57339 \dots - 0.28731 \dots ∣ = 1.86071 \dots

Δ u_{4} = 1.86071 \dots

u_{5} = - 0.36523 \dots : Δ u_{5} = 1.20816 \dots

f (u_{4}) = 2 (- 1.57339 \dots)^{2} + 0.56216 \dots (- 1.57339 \dots) + 2.85784 \dots = 6.92448 \dots

f^{^{'}} (u_{4}) = 4 (- 1.57339 \dots) + 0.56216 \dots = - 5.73142 \dots

u_{5} = - 0.36523 \dots

Δ u_{5} = ∣ - 0.36523 \dots - (- 1.57339 \dots) ∣ = 1.20816 \dots

Δ u_{5} = 1.20816 \dots

u_{6} = 2.88287 \dots : Δ u_{6} = 3.24811 \dots

f (u_{5}) = 2 (- 0.36523 \dots)^{2} + 0.56216 \dots (- 0.36523 \dots) + 2.85784 \dots = 2.91931 \dots

f^{^{'}} (u_{5}) = 4 (- 0.36523 \dots) + 0.56216 \dots = - 0.89877 \dots

u_{6} = 2.88287 \dots

Δ u_{6} = ∣ 2.88287 \dots - (- 0.36523 \dots) ∣ = 3.24811 \dots

Δ u_{6} = 3.24811 \dots

u_{7} = 1.13812 \dots : Δ u_{7} = 1.74475 \dots

f (u_{6}) = 2 \cdot 2.88287 \dots^{2} + 0.56216 \dots \cdot 2.88287 \dots + 2.85784 \dots = 21.10048 \dots

f^{^{'}} (u_{6}) = 4 \cdot 2.88287 \dots + 0.56216 \dots = 12.09368 \dots

u_{7} = 1.13812 \dots

Δ u_{7} = ∣ 1.13812 \dots - 2.88287 \dots ∣ = 1.74475 \dots

Δ u_{7} = 1.74475 \dots

u_{8} = - 0.05223 \dots : Δ u_{8} = 1.19036 \dots

f (u_{7}) = 2 \cdot 1.13812 \dots^{2} + 0.56216 \dots \cdot 1.13812 \dots + 2.85784 \dots = 6.08832 \dots

f^{^{'}} (u_{7}) = 4 \cdot 1.13812 \dots + 0.56216 \dots = 5.11467 \dots

u_{8} = - 0.05223 \dots

Δ u_{8} = ∣ - 0.05223 \dots - 1.13812 \dots ∣ = 1.19036 \dots

Δ u_{8} = 1.19036 \dots

u_{9} = - 8.07548 \dots : Δ u_{9} = 8.02325 \dots

f (u_{8}) = 2 (- 0.05223 \dots)^{2} + 0.56216 \dots (- 0.05223 \dots) + 2.85784 \dots = 2.83393 \dots

f^{^{'}} (u_{8}) = 4 (- 0.05223 \dots) + 0.56216 \dots = 0.35321 \dots

u_{9} = - 8.07548 \dots

Δ u_{9} = ∣ - 8.07548 \dots - (- 0.05223 \dots) ∣ = 8.02325 \dots

Δ u_{9} = 8.02325 \dots

u_{10} = - 4.01921 \dots : Δ u_{10} = 4.05626 \dots

f (u_{9}) = 2 (- 8.07548 \dots)^{2} + 0.56216 \dots (- 8.07548 \dots) + 2.85784 \dots = 128.74508 \dots

f^{^{'}} (u_{9}) = 4 (- 8.07548 \dots) + 0.56216 \dots = - 31.73978 \dots

u_{10} = - 4.01921 \dots

Δ u_{10} = ∣ - 4.01921 \dots - (- 8.07548 \dots) ∣ = 4.05626 \dots

Δ u_{10} = 4.05626 \dots

u_{11} = - 1.89822 \dots : Δ u_{11} = 2.12099 \dots

f (u_{10}) = 2 (- 4.01921 \dots)^{2} + 0.56216 \dots (- 4.01921 \dots) + 2.85784 \dots = 32.90662 \dots

f^{^{'}} (u_{10}) = 4 (- 4.01921 \dots) + 0.56216 \dots = - 15.51471 \dots

u_{11} = - 1.89822 \dots

Δ u_{11} = ∣ - 1.89822 \dots - (- 4.01921 \dots) ∣ = 2.12099 \dots

Δ u_{11} = 2.12099 \dots

u_{12} = - 0.61852 \dots : Δ u_{12} = 1.27970 \dots

f (u_{11}) = 2 (- 1.89822 \dots)^{2} + 0.56216 \dots (- 1.89822 \dots) + 2.85784 \dots = 8.99723 \dots

f^{^{'}} (u_{11}) = 4 (- 1.89822 \dots) + 0.56216 \dots = - 7.03073 \dots

u_{12} = - 0.61852 \dots

Δ u_{12} = ∣ - 0.61852 \dots - (- 1.89822 \dots) ∣ = 1.27970 \dots

Δ u_{12} = 1.27970 \dots

u_{13} = 1.09455 \dots : Δ u_{13} = 1.71307 \dots

f (u_{12}) = 2 (- 0.61852 \dots)^{2} + 0.56216 \dots (- 0.61852 \dots) + 2.85784 \dots = 3.27527 \dots

f^{^{'}} (u_{12}) = 4 (- 0.61852 \dots) + 0.56216 \dots = - 1.91192 \dots

u_{13} = 1.09455 \dots

Δ u_{13} = ∣ 1.09455 \dots - (- 0.61852 \dots) ∣ = 1.71307 \dots

Δ u_{13} = 1.71307 \dots

Cannot find solution

The solutions are

u \approx - 0.46746 \dots, u \approx 0.74854 \dots

Substitute back

u = cos (x)

cos (x) \approx - 0.46746 \dots, cos (x) \approx 0.74854 \dots

cos (x) \approx - 0.46746 \dots, cos (x) \approx 0.74854 \dots

cos (x) = - 0.46746 \dots : x = arccos (- 0.46746 \dots) + 2 πn, x = - arccos (- 0.46746 \dots) + 2 πn

cos (x) = - 0.46746 \dots

Apply trig inverse properties

cos (x) = - 0.46746 \dots

General solutions for

cos (x) = - 0.46746 \dots

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

x = arccos (- 0.46746 \dots) + 2 πn, x = - arccos (- 0.46746 \dots) + 2 πn

x = arccos (- 0.46746 \dots) + 2 πn, x = - arccos (- 0.46746 \dots) + 2 πn

cos (x) = 0.74854 \dots : x = arccos (0.74854 \dots) + 2 πn, x = 2 π - arccos (0.74854 \dots) + 2 πn

cos (x) = 0.74854 \dots

Apply trig inverse properties

cos (x) = 0.74854 \dots

General solutions for

cos (x) = 0.74854 \dots

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

x = arccos (0.74854 \dots) + 2 πn, x = 2 π - arccos (0.74854 \dots) + 2 πn

x = arccos (0.74854 \dots) + 2 πn, x = 2 π - arccos (0.74854 \dots) + 2 πn

Combine all the solutions

x = arccos (- 0.46746 \dots) + 2 πn, x = - arccos (- 0.46746 \dots) + 2 πn, x = arccos (0.74854 \dots) + 2 πn, x = 2 π - arccos (0.74854 \dots) + 2 πn

Show solutions in decimal form

x = 2.05721 \dots + 2 πn, x = - 2.05721 \dots + 2 πn, x = 0.72493 \dots + 2 πn, x = 2 π - 0.72493 \dots + 2 πn

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