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شائع علم المثلثات >

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

الحلّ

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

الحلّ

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

درجات

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

خطوات الحلّ

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

من الطرفين

cos (x)

اطرح

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)

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

:فعّل نطريّة فيتاغوروس

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

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

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

بسّط:

- 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))

افتح أقواس

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

فعّل قوانين سالب-موجب

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

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

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

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

بسّط:

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

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

جمّع التعابير المتشابهة

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

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

اجمع العناصر المتشابهة

= - 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

بالاستعانة بطريقة التعويض

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

cos (x) = u :

على افتراض أنّ

- 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

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

اكتب بالصورة الاعتياديّة

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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \approx - 0.46746 \dots

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

8 u^{3} + 4 u - 1

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

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

:استعمل قانون الجمع

= \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})

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 8 u^{3}

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

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

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 4 u

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

\frac{d u}{d u}

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

:استعمل المشتقة الأساسية

= 1

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

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

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

:مشتقة الثابت

= 0

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

بسّط

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

u_{0} = - 1

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

فعّل القسمة الطويلة:

\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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \approx 0.74854 \dots

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

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)

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

:استعمل قانون الجمع

= \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})

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 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})

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 1.86984 \dots u

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

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

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

:استخرج الثابت

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

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

:استعمل المشتقة الأساسية

= 2.43703 \dots \cdot 1

بسّط

= 2.43703 \dots

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

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

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

:مشتقة الثابت

= 0

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

بسّط

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

u_{0} = 1

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

فعّل القسمة الطويلة:

\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

بطريقة نيوتون ريبسون

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

جدّ حلًا لـ:

u \in R

لا يوجد حلّ لـ

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

تعريف تقريب نيوتن-ريبسون

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

f^{^{'}} (u)

جد:

4 u + 0.56216 \dots

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

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

:استعمل قانون الجمع

= \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})

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

:استخرج الثابت

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

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

:استعمل قانون الأسس

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

بسّط

= 4 u

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

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

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

:استخرج الثابت

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

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

:استعمل المشتقة الأساسية

= 0.56216 \dots \cdot 1

بسّط

= 0.56216 \dots

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

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

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

:مشتقة الثابت

= 0

= 4 u + 0.56216 \dots + 0

بسّط

= 4 u + 0.56216 \dots

u_{0} = - 5

استبدل

Δ u_{n + 1} < 0.000001

حتّى

u_{n + 1}

احسب

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

لا يمكن إيجاد حلّ

The solutions are

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

u = cos (x)

استبدل مجددًا

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

cos (x) = - 0.46746 \dots :

حلول عامّة لـ

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

cos (x) = 0.74854 \dots :

حلول عامّة لـ

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

وحّد الحلول

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

أظهر الحلّ بالتمثيل العشريّ

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

رسم

أمثلة شائعة

2sin(2x)=tan(2x)

2 sin (2 x) = tan (2 x)

cos(x)=0.22

cos (x) = 0.22

solvefor x,y=2arcsin(x-1)

so l v e f or x, y = 2 arcsin (x - 1)

sin(θ)-0.2*cos(θ)=0.48979

sin (θ) - 0.2 \cdot cos (θ) = 0.48979

cot(x-2)=-(sqrt(3))/3

cot (x - 2) = - \frac{3}{3}