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cot^2(x)cos(x)=cos(x)

الحلّ

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

الحلّ

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

+1

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

خطوات الحلّ

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

من الطرفين

cos (x)

اطرح

cot^{2} (x) cos (x) - cos (x) = 0

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

حلل إلى عوامل:

cos (x) (cot (x) + 1) (cot (x) - 1)

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

cos (x)

قم باخراج العامل المشترك

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

cot^{2} (x) - 1

حلل إلى عوامل:

(cot (x) + 1) (cot (x) - 1)

cot^{2} (x) - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

= (cot (x) + 1) (cot (x) - 1)

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

cos (x) (cot (x) + 1) (cot (x) - 1) = 0

حلّ كل جزء على حدة

cos (x) = 0 or cot (x) + 1 = 0 or cot (x) - 1 = 0

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

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

انقل

1

إلى الجانب الأيمن

cot (x) + 1 = 0

من الطرفين

1

اطرح

cot (x) + 1 - 1 = 0 - 1

بسّط

cot (x) = - 1

cot (x) = - 1

cot (x) = - 1 :

حلول عامّة لـ

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

cot (x) - 1 = 0 : x = \frac{π}{4} + πn

cot (x) - 1 = 0

انقل

1

إلى الجانب الأيمن

cot (x) - 1 = 0

للطرفين

1

أضف

cot (x) - 1 + 1 = 0 + 1

بسّط

cot (x) = 1

cot (x) = 1

cot (x) = 1 :

حلول عامّة لـ

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

وحّد الحلول

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

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