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Popular Trigonometry >

cos(3x)cos(x)=2cos(2x)cos(x)

Solution

cos (3 x) cos (x) = 2 cos (2 x) cos (x)

Solution

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.04719 \dots + 2 πn, x = 2 π - 1.04719 \dots + 2 πn, x = 2.46670 \dots + 2 πn, x = - 2.46670 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 60.00000 \dots^{\circ} + 36 0^{\circ} n, x = 299.99999 \dots^{\circ} + 36 0^{\circ} n, x = 141.33171 \dots^{\circ} + 36 0^{\circ} n, x = - 141.33171 \dots^{\circ} + 36 0^{\circ} n

Solution steps

cos (3 x) cos (x) = 2 cos (2 x) cos (x)

Subtract

2 cos (2 x) cos (x)

from both sides

cos (3 x) cos (x) - 2 cos (2 x) cos (x) = 0

Rewrite using trig identities

cos (3 x) cos (x) - 2 cos (2 x) cos (x)

Use the Double Angle identity:

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

= cos (3 x) cos (x) - 2 (2 cos^{2} (x) - 1) cos (x)

cos (3 x) cos (x) - (- 1 + 2 cos^{2} (x)) \cdot 2 cos (x) = 0

Factor

cos (3 x) cos (x) - (- 1 + 2 cos^{2} (x)) \cdot 2 cos (x) : cos (x) (cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1))

cos (3 x) cos (x) - (- 1 + 2 cos^{2} (x)) \cdot 2 cos (x)

Factor out common term

cos (x)

= cos (x) (cos (3 x) - 2 (- 1 + cos^{2} (x) \cdot 2))

Factor

cos (3 x) - 2 (2 cos^{2} (x) - 1) : cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1)

cos (3 x) - 2 (- 1 + cos^{2} (x) \cdot 2)

Factor

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

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

Rewrite

2 cos^{2} (x) - 1

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

2 cos^{2} (x) - 1

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Rewrite

1

1^{2}

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

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (2 cos (x) + 1) (2 cos (x) - 1)

= cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1)

= cos (x) (cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1))

cos (x) (cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1)) = 0

Solving each part separately

cos (x) = 0 or cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1) = 0 : x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn, x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

cos (3 x) - 2 (2 cos (x) + 1) (2 cos (x) - 1) = 0

Rewrite using trig identities

cos (3 x) - (- 1 + cos (x) 2) (1 + cos (x) 2) \cdot 2

cos (3 x) = 4 cos^{3} (x) - 3 cos (x)

cos (3 x)

Rewrite using trig identities

cos (3 x)

Rewrite as

= cos (2 x + x)

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (2 x) cos (x) - sin (2 x) sin (x)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

Simplify

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x) : cos (x) cos (2 x) - 2 sin^{2} (x) cos (x)

cos (2 x) cos (x) - 2 sin (x) cos (x) sin (x)

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

2 sin (x) cos (x) sin (x)

Apply exponent rule:

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

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

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

Add the numbers:

1 + 1 = 2

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

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

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

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

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

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

Expand

cos (x) (2 cos^{2} (x) - 1) : 2 cos^{3} (x) - cos (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = cos (x), b = 2 cos^{2} (x), c = 1

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

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

Simplify

2 cos^{2} (x) cos (x) - 1 \cdot cos (x) : 2 cos^{3} (x) - cos (x)

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

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

1 \cdot cos (x) = cos (x)

1 cos (x)

Multiply:

1 \cdot cos (x) = cos (x)

= cos (x)

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

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

= 2 cos^{3} (x) - cos (x) - 2 (1 - cos^{2} (x)) cos (x)

Expand

- 2 cos (x) (1 - cos^{2} (x)) : - 2 cos (x) + 2 cos^{3} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 2 cos (x), b = 1, c = cos^{2} (x)

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

Apply minus-plus rules

- (- a) = a

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

Simplify

- 2 \cdot 1 \cdot cos (x) + 2 cos^{2} (x) cos (x) : - 2 cos (x) + 2 cos^{3} (x)

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

2 \cdot 1 \cdot cos (x) = 2 cos (x)

2 \cdot 1 cos (x)

Multiply the numbers:

2 \cdot 1 = 2

= 2 cos (x)

2 cos^{2} (x) cos (x) = 2 cos^{3} (x)

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

Apply exponent rule:

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

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

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

Add the numbers:

2 + 1 = 3

= 2 cos^{3} (x)

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

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

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

Simplify

2 cos^{3} (x) - cos (x) - 2 cos (x) + 2 cos^{3} (x) : 4 cos^{3} (x) - 3 cos (x)

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

Group like terms

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

Add similar elements:

2 cos^{3} (x) + 2 cos^{3} (x) = 4 cos^{3} (x)

= 4 cos^{3} (x) - cos (x) - 2 cos (x)

Add similar elements:

- cos (x) - 2 cos (x) = - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x) - 2 (- 1 + 2 cos (x)) (1 + 2 cos (x))

- 3 cos (x) + 4 cos^{3} (x) - (- 1 + cos (x) 2) (1 + cos (x) 2) \cdot 2 = 0

Solve by substitution

- 3 cos (x) + 4 cos^{3} (x) - (- 1 + cos (x) 2) (1 + cos (x) 2) \cdot 2 = 0

Let:

cos (x) = u

- 3 u + 4 u^{3} - (- 1 + u 2) (1 + u 2) \cdot 2 = 0

- 3 u + 4 u^{3} - (- 1 + u 2) (1 + u 2) \cdot 2 = 0 : u \approx 1.28077 \dots, u \approx 0.50000 \dots, u \approx - 0.78077 \dots

- 3 u + 4 u^{3} - (- 1 + u 2) (1 + u 2) \cdot 2 = 0

Expand

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

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

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

Expand

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

Expand

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

(- 1 + u 2) (1 + u 2)

Apply Difference of Two Squares Formula:

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

a = u 2, b = 1

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

= (2 u)^{2} - 1

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

(u 2)^{2}

Apply exponent rule:

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

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

(2)^{2} : 2

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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

= 2

= u^{2} \cdot 2

= 2 u^{2} - 1

= 2 u^{2} - 1

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

Multiply the numbers:

2 \cdot 2 = 4

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

Multiply the numbers:

2 \cdot 1 = 2

= - 4 u^{2} + 2

= - 4 u^{2} + 2

= - 4 u^{2} + 2

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

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

Write in the standard form

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

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

Find one solution for

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

using Newton-Raphson:

u \approx 1.28077 \dots

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

Newton-Raphson Approximation Definition

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

Find

f^{^{'}} (u) : 12 u^{2} - 8 u - 3

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

Apply the Sum/Difference Rule:

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

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

\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} (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} (3 u) = 3

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

Take the constant out:

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

= 3 \frac{d u}{d u}

Apply the common derivative:

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

= 3 \cdot 1

Simplify

= 3

\frac{d}{d u} (2) = 0

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

Derivative of a constant:

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

= 0

= 12 u^{2} - 8 u - 3 + 0

Simplify

= 12 u^{2} - 8 u - 3

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 2 : Δ u_{1} = 1

f (u_{0}) = 4 \cdot 1^{3} - 4 \cdot 1^{2} - 3 \cdot 1 + 2 = - 1

f^{^{'}} (u_{0}) = 12 \cdot 1^{2} - 8 \cdot 1 - 3 = 1

u_{1} = 2

Δ u_{1} = ∣ 2 - 1 ∣ = 1

Δ u_{1} = 1

u_{2} = 1.58620 \dots : Δ u_{2} = 0.41379 \dots

f (u_{1}) = 4 \cdot 2^{3} - 4 \cdot 2^{2} - 3 \cdot 2 + 2 = 12

f^{^{'}} (u_{1}) = 12 \cdot 2^{2} - 8 \cdot 2 - 3 = 29

u_{2} = 1.58620 \dots

Δ u_{2} = ∣ 1.58620 \dots - 2 ∣ = 0.41379 \dots

Δ u_{2} = 0.41379 \dots

u_{3} = 1.36962 \dots : Δ u_{3} = 0.21658 \dots

f (u_{2}) = 4 \cdot 1.58620 \dots^{3} - 4 \cdot 1.58620 \dots^{2} - 3 \cdot 1.58620 \dots + 2 = 3.14108 \dots

f^{^{'}} (u_{2}) = 12 \cdot 1.58620 \dots^{2} - 8 \cdot 1.58620 \dots - 3 = 14.50297 \dots

u_{3} = 1.36962 \dots

Δ u_{3} = ∣ 1.36962 \dots - 1.58620 \dots ∣ = 0.21658 \dots

Δ u_{3} = 0.21658 \dots

u_{4} = 1.29192 \dots : Δ u_{4} = 0.07769 \dots

f (u_{3}) = 4 \cdot 1.36962 \dots^{3} - 4 \cdot 1.36962 \dots^{2} - 3 \cdot 1.36962 \dots + 2 = 0.66459 \dots

f^{^{'}} (u_{3}) = 12 \cdot 1.36962 \dots^{2} - 8 \cdot 1.36962 \dots - 3 = 8.55345 \dots

u_{4} = 1.29192 \dots

Δ u_{4} = ∣ 1.29192 \dots - 1.36962 \dots ∣ = 0.07769 \dots

Δ u_{4} = 0.07769 \dots

u_{5} = 1.28098 \dots : Δ u_{5} = 0.01093 \dots

f (u_{4}) = 4 \cdot 1.29192 \dots^{3} - 4 \cdot 1.29192 \dots^{2} - 3 \cdot 1.29192 \dots + 2 = 0.07319 \dots

f^{^{'}} (u_{4}) = 12 \cdot 1.29192 \dots^{2} - 8 \cdot 1.29192 \dots - 3 = 6.69344 \dots

u_{5} = 1.28098 \dots

Δ u_{5} = ∣ 1.28098 \dots - 1.29192 \dots ∣ = 0.01093 \dots

Δ u_{5} = 0.01093 \dots

u_{6} = 1.28077 \dots : Δ u_{6} = 0.00021 \dots

f (u_{5}) = 4 \cdot 1.28098 \dots^{3} - 4 \cdot 1.28098 \dots^{2} - 3 \cdot 1.28098 \dots + 2 = 0.00137 \dots

f^{^{'}} (u_{5}) = 12 \cdot 1.28098 \dots^{2} - 8 \cdot 1.28098 \dots - 3 = 6.44328 \dots

u_{6} = 1.28077 \dots

Δ u_{6} = ∣ 1.28077 \dots - 1.28098 \dots ∣ = 0.00021 \dots

Δ u_{6} = 0.00021 \dots

u_{7} = 1.28077 \dots : Δ u_{7} = 7.99005 E - 8

f (u_{6}) = 4 \cdot 1.28077 \dots^{3} - 4 \cdot 1.28077 \dots^{2} - 3 \cdot 1.28077 \dots + 2 = 5.14435 E - 7

f^{^{'}} (u_{6}) = 12 \cdot 1.28077 \dots^{2} - 8 \cdot 1.28077 \dots - 3 = 6.43844 \dots

u_{7} = 1.28077 \dots

Δ u_{7} = ∣ 1.28077 \dots - 1.28077 \dots ∣ = 7.99005 E - 8

Δ u_{7} = 7.99005 E - 8

u \approx 1.28077 \dots

Apply long division:

\frac{4 u ^{3} - 4 u ^{2} - 3 u + 2}{u - 1.28077 \dots} = 4 u^{2} + 1.12310 \dots u - 1.56155 \dots

4 u^{2} + 1.12310 \dots u - 1.56155 \dots \approx 0

Find one solution for

4 u^{2} + 1.12310 \dots u - 1.56155 \dots = 0

using Newton-Raphson:

u \approx 0.50000 \dots

4 u^{2} + 1.12310 \dots u - 1.56155 \dots = 0

Newton-Raphson Approximation Definition

f (u) = 4 u^{2} + 1.12310 \dots u - 1.56155 \dots

Find

f^{^{'}} (u) : 8 u + 1.12310 \dots

\frac{d}{d u} (4 u^{2} + 1.12310 \dots u - 1.56155 \dots)

Apply the Sum/Difference Rule:

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

= \frac{d}{d u} (4 u^{2}) + \frac{d}{d u} (1.12310 \dots u) - \frac{d}{d u} (1.56155 \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} (1.12310 \dots u) = 1.12310 \dots

\frac{d}{d u} (1.12310 \dots u)

Take the constant out:

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

= 1.12310 \dots \frac{d u}{d u}

Apply the common derivative:

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

= 1.12310 \dots \cdot 1

Simplify

= 1.12310 \dots

\frac{d}{d u} (1.56155 \dots) = 0

\frac{d}{d u} (1.56155 \dots)

Derivative of a constant:

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

= 0

= 8 u + 1.12310 \dots - 0

Simplify

= 8 u + 1.12310 \dots

Let

u_{0} = 1

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.60961 \dots : Δ u_{1} = 0.39038 \dots

f (u_{0}) = 4 \cdot 1^{2} + 1.12310 \dots \cdot 1 - 1.56155 \dots = 3.56155 \dots

f^{^{'}} (u_{0}) = 8 \cdot 1 + 1.12310 \dots = 9.12310 \dots

u_{1} = 0.60961 \dots

Δ u_{1} = ∣ 0.60961 \dots - 1 ∣ = 0.39038 \dots

Δ u_{1} = 0.39038 \dots

u_{2} = 0.50800 \dots : Δ u_{2} = 0.10160 \dots

f (u_{1}) = 4 \cdot 0.60961 \dots^{2} + 1.12310 \dots \cdot 0.60961 \dots - 1.56155 \dots = 0.60961 \dots

f^{^{'}} (u_{1}) = 8 \cdot 0.60961 \dots + 1.12310 \dots = 6

u_{2} = 0.50800 \dots

Δ u_{2} = ∣ 0.50800 \dots - 0.60961 \dots ∣ = 0.10160 \dots

Δ u_{2} = 0.10160 \dots

u_{3} = 0.50004 \dots : Δ u_{3} = 0.00796 \dots

f (u_{2}) = 4 \cdot 0.50800 \dots^{2} + 1.12310 \dots \cdot 0.50800 \dots - 1.56155 \dots = 0.04129 \dots

f^{^{'}} (u_{2}) = 8 \cdot 0.50800 \dots + 1.12310 \dots = 5.18718 \dots

u_{3} = 0.50004 \dots

Δ u_{3} = ∣ 0.50004 \dots - 0.50800 \dots ∣ = 0.00796 \dots

Δ u_{3} = 0.00796 \dots

u_{4} = 0.50000 \dots : Δ u_{4} = 0.00004 \dots

f (u_{3}) = 4 \cdot 0.50004 \dots^{2} + 1.12310 \dots \cdot 0.50004 \dots - 1.56155 \dots = 0.00025 \dots

f^{^{'}} (u_{3}) = 8 \cdot 0.50004 \dots + 1.12310 \dots = 5.12350 \dots

u_{4} = 0.50000 \dots

Δ u_{4} = ∣ 0.50000 \dots - 0.50004 \dots ∣ = 0.00004 \dots

Δ u_{4} = 0.00004 \dots

u_{5} = 0.5 : Δ u_{5} = 1.91092 E - 9

f (u_{4}) = 4 \cdot 0.50000 \dots^{2} + 1.12310 \dots \cdot 0.50000 \dots - 1.56155 \dots = 9.78986 E - 9

f^{^{'}} (u_{4}) = 8 \cdot 0.50000 \dots + 1.12310 \dots = 5.12310 \dots

u_{5} = 0.5

Δ u_{5} = ∣ 0.5 - 0.50000 \dots ∣ = 1.91092 E - 9

Δ u_{5} = 1.91092 E - 9

u \approx 0.50000 \dots

Apply long division:

\frac{4 u ^{2} + 1.12310 \dots u - 1.56155 \dots}{u - 0.5} = 4 u + 3.12310 \dots

4 u + 3.12310 \dots \approx 0

u \approx - 0.78077 \dots

The solutions are

u \approx 1.28077 \dots, u \approx 0.50000 \dots, u \approx - 0.78077 \dots

Substitute back

u = cos (x)

cos (x) \approx 1.28077 \dots, cos (x) \approx 0.50000 \dots, cos (x) \approx - 0.78077 \dots

cos (x) \approx 1.28077 \dots, cos (x) \approx 0.50000 \dots, cos (x) \approx - 0.78077 \dots

cos (x) = 1.28077 \dots :

No Solution

cos (x) = 1.28077 \dots

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = 0.50000 \dots : x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn

cos (x) = 0.50000 \dots

Apply trig inverse properties

cos (x) = 0.50000 \dots

General solutions for

cos (x) = 0.50000 \dots

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

x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn

x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn

cos (x) = - 0.78077 \dots : x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

cos (x) = - 0.78077 \dots

Apply trig inverse properties

cos (x) = - 0.78077 \dots

General solutions for

cos (x) = - 0.78077 \dots

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

x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

Combine all the solutions

x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn, x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arccos (0.50000 \dots) + 2 πn, x = 2 π - arccos (0.50000 \dots) + 2 πn, x = arccos (- 0.78077 \dots) + 2 πn, x = - arccos (- 0.78077 \dots) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 1.04719 \dots + 2 πn, x = 2 π - 1.04719 \dots + 2 πn, x = 2.46670 \dots + 2 πn, x = - 2.46670 \dots + 2 πn

Graph

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