English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

شائع علم المثلثات >

6tan^2(x)-2cos^2(x)=cos(2x)

الحلّ

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

الحلّ

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

درجات

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

6 tan^{2} (x) - 2 cos^{2} (x) = cos (2 x)

من الطرفين

cos (2 x)

اطرح

6 tan^{2} (x) - 2 cos^{2} (x) - cos (2 x) = 0

sin, cos :

عبّر بواسطة

- cos (2 x) - 2 cos^{2} (x) + 6 tan^{2} (x)

tan (x) = \frac{sin ( x )}{cos ( x )}

:Use the basic trigonometric identity

= - cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

بسّط:

\frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

- cos (2 x) - 2 cos^{2} (x) + 6 (\frac{sin ( x )}{cos ( x )})^{2}

6 (\frac{sin ( x )}{cos ( x )})^{2} = \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

6 (\frac{sin ( x )}{cos ( x )})^{2}

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

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

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

:فعّل قانون القوى

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

= 6 \cdot \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

:اضرب كسور

= \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

= - cos (2 x) - 2 cos^{2} (x) + \frac{6 sin ^{2} ( x )}{cos ^{2} ( x )}

cos (2 x) = \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )}, 2 cos^{2} (x) = \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )}

:حوّل الأعداد لكسور

= - \frac{cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} - \frac{2 cos ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- cos ( 2 x ) cos ^{2} ( x ) - 2 cos ^{2} ( x ) cos ^{2} ( x ) + sin ^{2} ( x ) \cdot 6}{cos ^{2} ( x )}

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6 = - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

- cos (2 x) cos^{2} (x) - 2 cos^{2} (x) cos^{2} (x) + sin^{2} (x) \cdot 6

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

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

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

:فعّل قانون القوى

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

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

2 + 2 = 4 :

اجمع الأعداد

= 2 cos^{4} (x)

= - cos^{2} (x) cos (2 x) - 2 cos^{4} (x) + 6 sin^{2} (x)

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) cos ( 2 x ) - 2 cos ^{4} ( x ) + 6 sin ^{2} ( x )}{cos ^{2} ( x )}

\frac{- 2 cos ^{4} ( x ) + 6 sin ^{2} ( x ) - cos ( 2 x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x) = 0

Rewrite using trig identities

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos (2 x) cos^{2} (x)

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

:فعّل متطابقة الزاوية المضاعفة

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

بسّط:

- 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

- 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{2} (x) (cos^{2} (x) - sin^{2} (x))

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

وسٌع:

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

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

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

فعّل قوانين سالب-موجب

- (- a) = a

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

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

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

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

:فعّل قانون القوى

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

= cos^{2 + 2} (x)

2 + 2 = 4 :

اجمع الأعداد

= cos^{4} (x)

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

= - 2 cos^{4} (x) + 6 sin^{2} (x) - cos^{4} (x) + cos^{2} (x) sin^{2} (x)

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

اجمع العناصر المتشابهة

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

= - 3 cos^{4} (x) + 6 sin^{2} (x) + cos^{2} (x) sin^{2} (x)

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

:فعّل نطريّة فيتاغوروس

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

= - 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

بسّط:

- 4 cos^{4} (x) - 5 cos^{2} (x) + 6

- 3 cos^{4} (x) + 6 (1 - cos^{2} (x)) + cos^{2} (x) (1 - cos^{2} (x))

6 (1 - cos^{2} (x))

وسٌع:

6 - 6 cos^{2} (x)

6 (1 - cos^{2} (x))

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = 6, b = 1, c = cos^{2} (x)

= 6 \cdot 1 - 6 cos^{2} (x)

6 \cdot 1 = 6 :

اضرب الأعداد

= 6 - 6 cos^{2} (x)

= - 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) (1 - cos^{2} (x))

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

وسٌع:

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

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

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

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

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

بسّط:

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

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

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

1 \cdot cos^{2} (x)

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

اضرب

= cos^{2} (x)

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

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

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

:فعّل قانون القوى

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

= cos^{2 + 2} (x)

2 + 2 = 4 :

اجمع الأعداد

= cos^{4} (x)

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

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

= - 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x)

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x)

بسّط:

- 4 cos^{4} (x) - 5 cos^{2} (x) + 6

- 3 cos^{4} (x) + 6 - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x)

جمّع التعابير المتشابهة

= - 3 cos^{4} (x) - 6 cos^{2} (x) + cos^{2} (x) - cos^{4} (x) + 6

- 6 cos^{2} (x) + cos^{2} (x) = - 5 cos^{2} (x) :

اجمع العناصر المتشابهة

= - 3 cos^{4} (x) - 5 cos^{2} (x) - cos^{4} (x) + 6

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

اجمع العناصر المتشابهة

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

= - 4 cos^{4} (x) - 5 cos^{2} (x) + 6

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

بالاستعانة بطريقة التعويض

6 - 4 cos^{4} (x) - 5 cos^{2} (x) = 0

cos (x) = u :

على افتراض أنّ

6 - 4 u^{4} - 5 u^{2} = 0

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

6 - 4 u^{4} - 5 u^{2} = 0

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

اكتب بالصورة الاعتياديّة

- 4 u^{4} - 5 u^{2} + 6 = 0

v^{2} = u^{4}

وكذلك

v = u^{2}

اكتب المعادلة مجددًا، بحيث أنّ

- 4 v^{2} - 5 v + 6 = 0

- 4 v^{2} - 5 v + 6 = 0

حلّ:

v = - 2, v = \frac{3}{4}

- 4 v^{2} - 5 v + 6 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- 4 v^{2} - 5 v + 6 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 4, b = - 5, c = 6

لـ

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

v_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 4 ) \cdot 6}{2 ( - 4 )}

(- 5)^{2} - 4 (- 4) \cdot 6 = 11

(- 5)^{2} - 4 (- 4) \cdot 6

- (- a) = a

فعّل القانون

= (- 5)^{2} + 4 \cdot 4 \cdot 6

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 4 \cdot 6

4 \cdot 4 \cdot 6 = 96 :

اضرب الأعداد

= 5^{2} + 96

5^{2} = 25

= 25 + 96

25 + 96 = 121 :

اجمع الأعداد

= 121

121 = 1 1^{2} :

حلّل العدد لعوامله أوّليّة

= 1 1^{2}

n a^{n} = a

:فعْل قانون الجذور

1 1^{2} = 11

= 11

v_{1, 2} = \frac{- ( - 5 ) \pm 11}{2 ( - 4 )}

Separate the solutions

v_{1} = \frac{- ( - 5 ) + 11}{2 ( - 4 )}, v_{2} = \frac{- ( - 5 ) - 11}{2 ( - 4 )}

v = \frac{- ( - 5 ) + 11}{2 ( - 4 )} : - 2

\frac{- ( - 5 ) + 11}{2 ( - 4 )}

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

:احذف الأقواس

= \frac{5 + 11}{- 2 \cdot 4}

5 + 11 = 16 :

اجمع الأعداد

= \frac{16}{- 2 \cdot 4}

2 \cdot 4 = 8 :

اضرب الأعداد

= \frac{16}{- 8}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{16}{8}

\frac{16}{8} = 2 :

اقسم الأعداد

= - 2

v = \frac{- ( - 5 ) - 11}{2 ( - 4 )} : \frac{3}{4}

\frac{- ( - 5 ) - 11}{2 ( - 4 )}

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

:احذف الأقواس

= \frac{5 - 11}{- 2 \cdot 4}

5 - 11 = - 6 :

اطرح الأعداد

= \frac{- 6}{- 2 \cdot 4}

2 \cdot 4 = 8 :

اضرب الأعداد

= \frac{- 6}{- 8}

\frac{- a}{- b} = \frac{a}{b}

: استخدم ميزات الكسور التالية

= \frac{6}{8}

2 :

إلغ العوامل المشتركة

= \frac{3}{4}

حلول المعادلة التربيعيّة هي

v = - 2, v = \frac{3}{4}

v = - 2, v = \frac{3}{4}

Substitute back

v = u^{2},

solve for

u

u^{2} = - 2

حلّ:

u = 2 i, u = - 2 i

u^{2} = - 2

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

الحلول هي

x^{2} = f (a)

لـ

u = - 2, u = - - 2

- 2

بسّط:

2 i

- 2

- a = - 1 a

:فعْل قانون الجذور

- 2 = - 1 2

= - 1 2

- 1 = i

:فعّل قانون الأعداد التخيليّة

= 2 i

- - 2

بسّط:

- 2 i

- - 2

- 2

بسّط:

2 i

- 2

- a = - 1 a

:فعْل قانون الجذور

- 2 = - 1 2

= - 1 2

- 1 = i

:فعّل قانون الأعداد التخيليّة

= 2 i

= - 2 i

u = 2 i, u = - 2 i

u^{2} = \frac{3}{4}

حلّ:

u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

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

الحلول هي

x^{2} = f (a)

لـ

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

a \geq 0, b \geq 0

بافتراض أنّ

n \frac{a}{b} = \frac{n a}{n b}

:فعّل قانون الجذور

= \frac{3}{4}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

\frac{3}{4}

بسّط:

\frac{3}{2}

\frac{3}{4}

a \geq 0, b \geq 0

بافتراض أنّ

n \frac{a}{b} = \frac{n a}{n b}

:فعّل قانون الجذور

= \frac{3}{4}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

The solutions are

u = 2 i, u = - 2 i, u = \frac{3}{2}, u = - \frac{3}{2}

u = cos (x)

استبدل مجددًا

cos (x) = 2 i, cos (x) = - 2 i, cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = 2 i, cos (x) = - 2 i, cos (x) = \frac{3}{2}, cos (x) = - \frac{3}{2}

cos (x) = 2 i :

لا يوجد حلّ

cos (x) = 2 i

لا يوجد حلّ

cos (x) = - 2 i :

لا يوجد حلّ

cos (x) = - 2 i

لا يوجد حلّ

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2} :

حلول عامّة لـ

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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

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

cos (x) = - \frac{3}{2} :

حلول عامّة لـ

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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

وحّد الحلول

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

رسم

أمثلة شائعة

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

2 cos^{2} (x) - 3 sin (x) - 3 = 0

6tan(θ)+5=0

6 tan (θ) + 5 = 0

solvefor x,y=arcsin(x)

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

tan(a)= 5/12

tan (a) = \frac{5}{12}

2sin(x)-3=-csc(x)

2 sin (x) - 3 = - csc (x)