English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

sin^4(x)+cos^4(x)= 7/9

الحلّ

sin^{4} (x) + cos^{4} (x) = \frac{7}{9}

الحلّ

x = 0.36486 \dots + 2 πn, x = 2 π - 0.36486 \dots + 2 πn, x = 2.77672 \dots + 2 πn, x = - 2.77672 \dots + 2 πn, x = 1.20593 \dots + 2 πn, x = 2 π - 1.20593 \dots + 2 πn, x = 1.93566 \dots + 2 πn, x = - 1.93566 \dots + 2 πn

درجات

x = 20.90515 \dots^{\circ} + 36 0^{\circ} n, x = 339.09484 \dots^{\circ} + 36 0^{\circ} n, x = 159.09484 \dots^{\circ} + 36 0^{\circ} n, x = - 159.09484 \dots^{\circ} + 36 0^{\circ} n, x = 69.09484 \dots^{\circ} + 36 0^{\circ} n, x = 290.90515 \dots^{\circ} + 36 0^{\circ} n, x = 110.90515 \dots^{\circ} + 36 0^{\circ} n, x = - 110.90515 \dots^{\circ} + 36 0^{\circ} n

خطوات الحلّ

sin^{4} (x) + cos^{4} (x) = \frac{7}{9}

من الطرفين

\frac{7}{9}

اطرح

sin^{4} (x) + cos^{4} (x) - \frac{7}{9} = 0

sin^{4} (x) + cos^{4} (x) - \frac{7}{9}

بسّط:

\frac{9 sin ^{4} ( x ) + 9 cos ^{4} ( x ) - 7}{9}

sin^{4} (x) + cos^{4} (x) - \frac{7}{9}

sin^{4} (x) = \frac{sin ^{4} ( x ) 9}{9}, cos^{4} (x) = \frac{cos ^{4} ( x ) 9}{9}

:حوّل الأعداد لكسور

= \frac{sin ^{4} ( x ) \cdot 9}{9} + \frac{cos ^{4} ( x ) \cdot 9}{9} - \frac{7}{9}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{sin ^{4} ( x ) \cdot 9 + cos ^{4} ( x ) \cdot 9 - 7}{9}

\frac{9 sin ^{4} ( x ) + 9 cos ^{4} ( x ) - 7}{9} = 0

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

9 sin^{4} (x) + 9 cos^{4} (x) - 7 = 0

a^{b} = a^{2} a^{b - 2} :

فعّل قانون القوى

- 7 + 9 cos^{4} (x) + 9 sin^{2} (x) sin^{2} (x) = 0

Rewrite using trig identities

- 7 + 9 cos^{4} (x) + 9 sin^{2} (x) sin^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= - 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

- 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

بسّط:

18 cos^{4} (x) - 18 cos^{2} (x) + 2

- 7 + 9 cos^{4} (x) + 9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

9 (1 - cos^{2} (x)) (1 - cos^{2} (x)) = 9 (1 - cos^{2} (x))^{2}

9 (1 - cos^{2} (x)) (1 - cos^{2} (x))

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

:فعّل قانون القوى

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

= 9 (1 - cos^{2} (x))^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= 9 (1 - cos^{2} (x))^{2}

= - 7 + 9 cos^{4} (x) + 9 (- cos^{2} (x) + 1)^{2}

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

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

:فعّل صيغة الضرب المختصر

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

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

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

بسّط:

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

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

2 \cdot 1 = 2 :

اضرب الأعداد

= 2 cos^{2} (x)

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

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

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

:فعّل قانون القوى

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

2 \cdot 2 = 4 :

اضرب الأعداد

= cos^{4} (x)

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

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

= - 7 + 9 cos^{4} (x) + 9 (1 - 2 cos^{2} (x) + cos^{4} (x))

9 (1 - 2 cos^{2} (x) + cos^{4} (x))

وسٌع:

9 - 18 cos^{2} (x) + 9 cos^{4} (x)

9 (1 - 2 cos^{2} (x) + cos^{4} (x))

فعّل قانون ضرب الأقواس

= 9 \cdot 1 + 9 (- 2 cos^{2} (x)) + 9 cos^{4} (x)

فعّل قوانين سالب-موجب

+ (- a) = - a

= 9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

بسّط:

9 - 18 cos^{2} (x) + 9 cos^{4} (x)

9 \cdot 1 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

9 \cdot 1 = 9 :

اضرب الأعداد

= 9 - 9 \cdot 2 cos^{2} (x) + 9 cos^{4} (x)

9 \cdot 2 = 18 :

اضرب الأعداد

= 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

= 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

= - 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

- 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

بسّط:

18 cos^{4} (x) - 18 cos^{2} (x) + 2

- 7 + 9 cos^{4} (x) + 9 - 18 cos^{2} (x) + 9 cos^{4} (x)

جمّع التعابير المتشابهة

= 9 cos^{4} (x) - 18 cos^{2} (x) + 9 cos^{4} (x) - 7 + 9

9 cos^{4} (x) + 9 cos^{4} (x) = 18 cos^{4} (x) :

اجمع العناصر المتشابهة

= 18 cos^{4} (x) - 18 cos^{2} (x) - 7 + 9

- 7 + 9 = 2 :

اطرح/اجمع الأعداد

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

= 18 cos^{4} (x) - 18 cos^{2} (x) + 2

2 - 18 cos^{2} (x) + 18 cos^{4} (x) = 0

بالاستعانة بطريقة التعويض

2 - 18 cos^{2} (x) + 18 cos^{4} (x) = 0

cos (x) = u :

على افتراض أنّ

2 - 18 u^{2} + 18 u^{4} = 0

2 - 18 u^{2} + 18 u^{4} = 0 : u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}, u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

2 - 18 u^{2} + 18 u^{4} = 0

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

اكتب بالصورة الاعتياديّة

18 u^{4} - 18 u^{2} + 2 = 0

v^{2} = u^{4}

وكذلك

v = u^{2}

اكتب المعادلة مجددًا، بحيث أنّ

18 v^{2} - 18 v + 2 = 0

18 v^{2} - 18 v + 2 = 0

حلّ:

v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

18 v^{2} - 18 v + 2 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

18 v^{2} - 18 v + 2 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 18, b = - 18, c = 2

لـ

v_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 18 \cdot 2}{2 \cdot 18}

v_{1, 2} = \frac{- ( - 18 ) \pm ( - 18 ) ^{2} - 4 \cdot 18 \cdot 2}{2 \cdot 18}

(- 18)^{2} - 4 \cdot 18 \cdot 2 = 65

(- 18)^{2} - 4 \cdot 18 \cdot 2

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 18)^{2} = 1 8^{2}

= 1 8^{2} - 4 \cdot 18 \cdot 2

4 \cdot 18 \cdot 2 = 144 :

اضرب الأعداد

= 1 8^{2} - 144

1 8^{2} = 324

= 324 - 144

324 - 144 = 180 :

اطرح الأعداد

= 180

180

تحليل لعوامل أوّليّة لـ:

2^{2} \cdot 3^{2} \cdot 5

180

180 = 90 \cdot 2, 2

ينقسم على

180

= 2 \cdot 90

90 = 45 \cdot 2, 2

ينقسم على

90

= 2 \cdot 2 \cdot 45

45 = 15 \cdot 3, 3

ينقسم على

45

= 2 \cdot 2 \cdot 3 \cdot 15

15 = 5 \cdot 3, 3

ينقسم على

15

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

مركّب من أعداد أوّليّة فقط، لذلك تحليل إضافيّ غير ممكن

2, 3, 5

= 2 \cdot 2 \cdot 3 \cdot 3 \cdot 5

= 2^{2} \cdot 3^{2} \cdot 5

= 2^{2} \cdot 3^{2} \cdot 5

n ab = n a n b

:فعْل قانون الجذور

= 5 2^{2} 3^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 25 3^{2}

n a^{n} = a

:فعْل قانون الجذور

3^{2} = 3

= 2 \cdot 35

بسّط

= 65

v_{1, 2} = \frac{- ( - 18 ) \pm 6 5}{2 \cdot 18}

Separate the solutions

v_{1} = \frac{- ( - 18 ) + 6 5}{2 \cdot 18}, v_{2} = \frac{- ( - 18 ) - 6 5}{2 \cdot 18}

v = \frac{- ( - 18 ) + 6 5}{2 \cdot 18} : \frac{3 + 5}{6}

\frac{- ( - 18 ) + 6 5}{2 \cdot 18}

- (- a) = a

فعّل القانون

= \frac{18 + 6 5}{2 \cdot 18}

2 \cdot 18 = 36 :

اضرب الأعداد

= \frac{18 + 6 5}{36}

18 + 65

حلل إلى عوامل:

6 (3 + 5)

18 + 65

أعد الكتابة كـ

= 6 \cdot 3 + 65

6

قم باخراج العامل المشترك

= 6 (3 + 5)

= \frac{6 ( 3 + 5 )}{36}

6 :

إلغ العوامل المشتركة

= \frac{3 + 5}{6}

v = \frac{- ( - 18 ) - 6 5}{2 \cdot 18} : \frac{3 - 5}{6}

\frac{- ( - 18 ) - 6 5}{2 \cdot 18}

- (- a) = a

فعّل القانون

= \frac{18 - 6 5}{2 \cdot 18}

2 \cdot 18 = 36 :

اضرب الأعداد

= \frac{18 - 6 5}{36}

18 - 65

حلل إلى عوامل:

6 (3 - 5)

18 - 65

أعد الكتابة كـ

= 6 \cdot 3 - 65

6

قم باخراج العامل المشترك

= 6 (3 - 5)

= \frac{6 ( 3 - 5 )}{36}

6 :

إلغ العوامل المشتركة

= \frac{3 - 5}{6}

حلول المعادلة التربيعيّة هي

v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

v = \frac{3 + 5}{6}, v = \frac{3 - 5}{6}

Substitute back

v = u^{2},

solve for

u

u^{2} = \frac{3 + 5}{6}

حلّ:

u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}

u^{2} = \frac{3 + 5}{6}

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}

u^{2} = \frac{3 - 5}{6}

حلّ:

u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

u^{2} = \frac{3 - 5}{6}

x = f (a), - f (a)

الحلول هي

x^{2} = f (a)

لـ

u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

The solutions are

u = \frac{3 + 5}{6}, u = - \frac{3 + 5}{6}, u = \frac{3 - 5}{6}, u = - \frac{3 - 5}{6}

u = cos (x)

استبدل مجددًا

cos (x) = \frac{3 + 5}{6}, cos (x) = - \frac{3 + 5}{6}, cos (x) = \frac{3 - 5}{6}, cos (x) = - \frac{3 - 5}{6}

cos (x) = \frac{3 + 5}{6}, cos (x) = - \frac{3 + 5}{6}, cos (x) = \frac{3 - 5}{6}, cos (x) = - \frac{3 - 5}{6}

cos (x) = \frac{3 + 5}{6} : x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

cos (x) = \frac{3 + 5}{6}

Apply trig inverse properties

cos (x) = \frac{3 + 5}{6}

cos (x) = \frac{3 + 5}{6} :

حلول عامّة لـ

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

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn

cos (x) = - \frac{3 + 5}{6} : x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

cos (x) = - \frac{3 + 5}{6}

Apply trig inverse properties

cos (x) = - \frac{3 + 5}{6}

cos (x) = - \frac{3 + 5}{6} :

حلول عامّة لـ

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

x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn

cos (x) = \frac{3 - 5}{6} : x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

cos (x) = \frac{3 - 5}{6}

Apply trig inverse properties

cos (x) = \frac{3 - 5}{6}

cos (x) = \frac{3 - 5}{6} :

حلول عامّة لـ

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

x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn

cos (x) = - \frac{3 - 5}{6} : x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

cos (x) = - \frac{3 - 5}{6}

Apply trig inverse properties

cos (x) = - \frac{3 - 5}{6}

cos (x) = - \frac{3 - 5}{6} :

حلول عامّة لـ

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

x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

وحّد الحلول

x = arccos \frac{3 + 5}{6} + 2 πn, x = 2 π - arccos \frac{3 + 5}{6} + 2 πn, x = arccos - \frac{3 + 5}{6} + 2 πn, x = - arccos - \frac{3 + 5}{6} + 2 πn, x = arccos \frac{3 - 5}{6} + 2 πn, x = 2 π - arccos \frac{3 - 5}{6} + 2 πn, x = arccos - \frac{3 - 5}{6} + 2 πn, x = - arccos - \frac{3 - 5}{6} + 2 πn

أظهر الحلّ بالتمثيل العشريّ

x = 0.36486 \dots + 2 πn, x = 2 π - 0.36486 \dots + 2 πn, x = 2.77672 \dots + 2 πn, x = - 2.77672 \dots + 2 πn, x = 1.20593 \dots + 2 πn, x = 2 π - 1.20593 \dots + 2 πn, x = 1.93566 \dots + 2 πn, x = - 1.93566 \dots + 2 πn

رسم

أمثلة شائعة

tan(θ/5)+sqrt(3)=0

tan (\frac{θ}{5}) + 3 = 0

3tan(x)-sqrt(3)=0,0<= x<2pi

3 tan (x) - 3 = 0, 0 \leq x < 2 π

sec^2(θ)=2

sec^{2} (θ) = 2

sin^2(θ)=8-2cos^2(θ)

sin^{2} (θ) = 8 - 2 cos^{2} (θ)

10sin(x)+16=3sin(x)+9

10 sin (x) + 16 = 3 sin (x) + 9