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شائع علم المثلثات >

5cos^2(2x)+4cos^2(x)-5=0

الحلّ

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

الحلّ

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

درجات

x = 26.56505 \dots^{\circ} + 36 0^{\circ} n, x = 333.43494 \dots^{\circ} + 36 0^{\circ} n, x = 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = - 153.43494 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

Rewrite using trig identities

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

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

:فعّل متطابقة الزاوية المضاعفة

= - 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2}

- 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2}

بسّط:

20 cos^{4} (x) - 16 cos^{2} (x)

- 5 + 4 cos^{2} (x) + 5 (2 cos^{2} (x) - 1)^{2}

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

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

:فعّل صيغة الضرب المختصر

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

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

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

بسّط:

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

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

1^{a} = 1

فعّل القانون

1^{2} = 1

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

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

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

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

2 \cdot 2 = 4 :

اضرب الأعداد

= cos^{4} (x)

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

2^{2} = 4

= 4 cos^{4} (x)

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

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

2 \cdot 2 \cdot 1 = 4 :

اضرب الأعداد

= 4 cos^{2} (x)

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

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

= - 5 + 4 cos^{2} (x) + 5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1)

5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1)

وسٌع:

20 cos^{4} (x) - 20 cos^{2} (x) + 5

5 (4 cos^{4} (x) - 4 cos^{2} (x) + 1)

فعّل قانون ضرب الأقواس

= 5 \cdot 4 cos^{4} (x) + 5 (- 4 cos^{2} (x)) + 5 \cdot 1

فعّل قوانين سالب-موجب

+ (- a) = - a

= 5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1

5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1

بسّط:

20 cos^{4} (x) - 20 cos^{2} (x) + 5

5 \cdot 4 cos^{4} (x) - 5 \cdot 4 cos^{2} (x) + 5 \cdot 1

5 \cdot 4 = 20 :

اضرب الأعداد

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5 \cdot 1

5 \cdot 1 = 5 :

اضرب الأعداد

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5

= 20 cos^{4} (x) - 20 cos^{2} (x) + 5

= - 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5

- 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5

بسّط:

20 cos^{4} (x) - 16 cos^{2} (x)

- 5 + 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) + 5

جمّع التعابير المتشابهة

= 4 cos^{2} (x) + 20 cos^{4} (x) - 20 cos^{2} (x) - 5 + 5

4 cos^{2} (x) - 20 cos^{2} (x) = - 16 cos^{2} (x) :

اجمع العناصر المتشابهة

= - 16 cos^{2} (x) + 20 cos^{4} (x) - 5 + 5

- 5 + 5 = 0

= 20 cos^{4} (x) - 16 cos^{2} (x)

= 20 cos^{4} (x) - 16 cos^{2} (x)

= 20 cos^{4} (x) - 16 cos^{2} (x)

- 16 cos^{2} (x) + 20 cos^{4} (x) = 0

بالاستعانة بطريقة التعويض

- 16 cos^{2} (x) + 20 cos^{4} (x) = 0

cos (x) = u :

على افتراض أنّ

- 16 u^{2} + 20 u^{4} = 0

- 16 u^{2} + 20 u^{4} = 0 : u = \frac{2 5}{5}, u = - \frac{2 5}{5}, u = 0

- 16 u^{2} + 20 u^{4} = 0

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

اكتب بالصورة الاعتياديّة

20 u^{4} - 16 u^{2} = 0

v^{2} = u^{4}

وكذلك

v = u^{2}

اكتب المعادلة مجددًا، بحيث أنّ

20 v^{2} - 16 v = 0

20 v^{2} - 16 v = 0

حلّ:

v = \frac{4}{5}, v = 0

20 v^{2} - 16 v = 0

حلّ بالاستعانة بالصيغة التربيعيّة

20 v^{2} - 16 v = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 20, b = - 16, c = 0

لـ

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 20 \cdot 0}{2 \cdot 20}

v_{1, 2} = \frac{- ( - 16 ) \pm ( - 16 ) ^{2} - 4 \cdot 20 \cdot 0}{2 \cdot 20}

(- 16)^{2} - 4 \cdot 20 \cdot 0 = 16

(- 16)^{2} - 4 \cdot 20 \cdot 0

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 16)^{2} = 1 6^{2}

= 1 6^{2} - 4 \cdot 20 \cdot 0

0 \cdot a = 0

فعّل القانون

= 1 6^{2} - 0

1 6^{2} - 0 = 1 6^{2}

= 1 6^{2}

a \geq 0

بافتراض أنّ

n a^{n} = a

:فعّل قانون الجذور

= 16

v_{1, 2} = \frac{- ( - 16 ) \pm 16}{2 \cdot 20}

Separate the solutions

v_{1} = \frac{- ( - 16 ) + 16}{2 \cdot 20}, v_{2} = \frac{- ( - 16 ) - 16}{2 \cdot 20}

v = \frac{- ( - 16 ) + 16}{2 \cdot 20} : \frac{4}{5}

\frac{- ( - 16 ) + 16}{2 \cdot 20}

- (- a) = a

فعّل القانون

= \frac{16 + 16}{2 \cdot 20}

16 + 16 = 32 :

اجمع الأعداد

= \frac{32}{2 \cdot 20}

2 \cdot 20 = 40 :

اضرب الأعداد

= \frac{32}{40}

8 :

إلغ العوامل المشتركة

= \frac{4}{5}

v = \frac{- ( - 16 ) - 16}{2 \cdot 20} : 0

\frac{- ( - 16 ) - 16}{2 \cdot 20}

- (- a) = a

فعّل القانون

= \frac{16 - 16}{2 \cdot 20}

16 - 16 = 0 :

اطرح الأعداد

= \frac{0}{2 \cdot 20}

2 \cdot 20 = 40 :

اضرب الأعداد

= \frac{0}{40}

\frac{0}{a} = 0, a \neq = 0

فعّل القانون

= 0

حلول المعادلة التربيعيّة هي

v = \frac{4}{5}, v = 0

v = \frac{4}{5}, v = 0

Substitute back

v = u^{2},

solve for

u

u^{2} = \frac{4}{5}

حلّ:

u = \frac{2 5}{5}, u = - \frac{2 5}{5}

u^{2} = \frac{4}{5}

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

الحلول هي

x^{2} = f (a)

لـ

u = \frac{4}{5}, u = - \frac{4}{5}

\frac{4}{5} = \frac{2 5}{5}

\frac{4}{5}

a \geq 0, b \geq 0

بافتراض أنّ

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

:فعّل قانون الجذور

= \frac{4}{5}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{2}{5}

\frac{2}{5}

حوّل لصيغة عدد كسريّ:

\frac{2 5}{5}

\frac{2}{5}

\frac{5}{5}

اضرب بالمرافق

= \frac{2 5}{5 5}

55 = 5

55

a a = a

:فعْل قانون الجذور

55 = 5

= 5

= \frac{2 5}{5}

= \frac{2 5}{5}

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

- \frac{4}{5}

\frac{4}{5}

بسّط:

\frac{2}{5}

\frac{4}{5}

a \geq 0, b \geq 0

بافتراض أنّ

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

:فعّل قانون الجذور

= \frac{4}{5}

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= \frac{2}{5}

= - \frac{2}{5}

- \frac{2}{5}

حوّل لصيغة عدد كسريّ:

- \frac{2 5}{5}

- \frac{2}{5}

\frac{5}{5}

اضرب بالمرافق

= - \frac{2 5}{5 5}

55 = 5

55

a a = a

:فعْل قانون الجذور

55 = 5

= 5

= - \frac{2 5}{5}

= - \frac{2 5}{5}

u = \frac{2 5}{5}, u = - \frac{2 5}{5}

u^{2} = 0

حلّ:

u = 0

u^{2} = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

u = 0

The solutions are

u = \frac{2 5}{5}, u = - \frac{2 5}{5}, u = 0

u = cos (x)

استبدل مجددًا

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}, cos (x) = 0

cos (x) = \frac{2 5}{5}, cos (x) = - \frac{2 5}{5}, cos (x) = 0

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

cos (x) = \frac{2 5}{5}

Apply trig inverse properties

cos (x) = \frac{2 5}{5}

cos (x) = \frac{2 5}{5} :

حلول عامّة لـ

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

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

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

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

cos (x) = - \frac{2 5}{5}

Apply trig inverse properties

cos (x) = - \frac{2 5}{5}

cos (x) = - \frac{2 5}{5} :

حلول عامّة لـ

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

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

x = arccos (- \frac{2 5}{5}) + 2 πn, x = - arccos (- \frac{2 5}{5}) + 2 πn

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

cos (x) = 0

cos (x) = 0 :

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

وحّد الحلول

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

أظهر الحلّ بالتمثيل العشريّ

x = 0.46364 \dots + 2 πn, x = 2 π - 0.46364 \dots + 2 πn, x = 2.67794 \dots + 2 πn, x = - 2.67794 \dots + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

رسم

أمثلة شائعة

sin(3x+10)=cos(x+24)

sin (3 x + 10) = cos (x + 2 4^{\circ})

tan(x^2)+1=0

tan (x^{2}) + 1 = 0

((1+cos^2(a)))/(sin^2(a))= 5/3

\frac{( 1 + cos ^{2} ( a ) )}{sin ^{2} ( a )} = \frac{5}{3}

d^2+13d+36=(sin^2(x))/2

d^{2} + 13 d + 36 = \frac{sin ^{2} ( x )}{2}

(tan^2(x)-4)/(cos(x)+5)=0

\frac{tan ^{2} ( x ) - 4}{cos ( x ) + 5} = 0