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(cot(a))/(cos(a))=cos(a)

الحلّ

\frac{cot ( a )}{cos ( a )} = cos (a)

الحلّ

a \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{cot ( a )}{cos ( a )} = cos (a)

من الطرفين

cos (a)

اطرح

\frac{cot ( a )}{cos ( a )} - cos (a) = 0

\frac{cot ( a )}{cos ( a )} - cos (a)

بسّط:

\frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )}

\frac{cot ( a )}{cos ( a )} - cos (a)

cos (a) = \frac{cos ( a ) cos ( a )}{cos ( a )}

:حوّل الأعداد لكسور

= \frac{cot ( a )}{cos ( a )} - \frac{cos ( a ) cos ( a )}{cos ( a )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{cot ( a ) - cos ( a ) cos ( a )}{cos ( a )}

cot (a) - cos (a) cos (a) = cot (a) - cos^{2} (a)

cot (a) - cos (a) cos (a)

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

cos (a) cos (a)

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

:فعّل قانون القوى

cos (a) cos (a) = cos^{1 + 1} (a)

= cos^{1 + 1} (a)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (a)

= cot (a) - cos^{2} (a)

= \frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )}

\frac{cot ( a ) - cos ^{2} ( a )}{cos ( a )} = 0

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

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

sin, cos :

عبّر بواسطة

- cos^{2} (a) + cot (a)

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

:Use the basic trigonometric identity

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

\frac{cos ( a ) - cos ^{2} ( a ) sin ( a )}{sin ( a )} = 0

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

cos (a) - cos^{2} (a) sin (a) = 0

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

حلل إلى عوامل:

cos (a) (1 - sin (a) cos (a))

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

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

:فعّل قانون القوى

sin (a) cos^{2} (a) = cos (a) cos (a)

= cos (a) - cos (a) cos (a)

cos (a)

قم باخراج العامل المشترك

= cos (a) (1 - sin (a) cos (a))

cos (a) (1 - sin (a) cos (a)) = 0

حلّ كل جزء على حدة

cos (a) = 0 or 1 - sin (a) cos (a) = 0

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

cos (a) = 0

cos (a) = 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}

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

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

1 - sin (a) cos (a) = 0 :

لا يوجد حلّ

1 - sin (a) cos (a) = 0

Rewrite using trig identities

1 - sin (a) cos (a)

2 sin (x) cos (x) = sin (2 x)

:فعّل متطابقة الزاوية المضاعفة

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

= 1 - \frac{sin ( 2 a )}{2}

1 - \frac{sin ( 2 a )}{2} = 0

انقل

1

إلى الجانب الأيمن

1 - \frac{sin ( 2 a )}{2} = 0

من الطرفين

1

اطرح

1 - \frac{sin ( 2 a )}{2} - 1 = 0 - 1

بسّط

- \frac{sin ( 2 a )}{2} = - 1

- \frac{sin ( 2 a )}{2} = - 1

2

اضرب الطرفين بـ

- \frac{sin ( 2 a )}{2} = - 1

2

اضرب الطرفين بـ

2 (- \frac{sin ( 2 a )}{2}) = 2 (- 1)

بسّط

- sin (2 a) = - 2

- sin (2 a) = - 2

- 1

اقسم الطرفين على

- sin (2 a) = - 2

- 1

اقسم الطرفين على

\frac{- sin ( 2 a )}{- 1} = \frac{- 2}{- 1}

بسّط

sin (2 a) = 2

sin (2 a) = 2

- 1 \leq sin (x) \leq 1

لا يوجد حلّ

وحّد الحلول

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

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

a \in R لا يوجد حلّ لـ

أمثلة شائعة

cos(x)-2=0

cos (x) - 2 = 0

tan^2(x)+3=3sec(x)

tan^{2} (x) + 3 = 3 sec (x)

sqrt(3)cot^2(x)-4cot(x)+sqrt(3)=0

3 cot^{2} (x) - 4 cot (x) + 3 = 0

cot(θ)+tan(θ)=2

cot (θ) + tan (θ) = 2

3cos^2(θ)-1=0

3 cos^{2} (θ) - 1 = 0