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شائع علم المثلثات >

((1+cot^2(x)))/(cos^2(x))=cot^2(x)

الحلّ

\frac{( 1 + cot ^{2} ( x ) )}{cos ^{2} ( x )} = cot^{2} (x)

الحلّ

x \in R لا يوجد حلّ لـ

خطوات الحلّ

\frac{( 1 + cot ^{2} ( x ) )}{cos ^{2} ( x )} = cot^{2} (x)

من الطرفين

cot^{2} (x)

اطرح

\frac{1 + cot ^{2} ( x )}{cos ^{2} ( x )} - cot^{2} (x) = 0

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{1 + cot ^{2} ( x ) - cot ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} = 0

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

= - cos^{2} (x) cot^{2} (x) + csc^{2} (x)

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

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

حلل إلى عوامل:

(csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x))

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

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

كـ

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

اكتب مجددًا

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

a^{m} b^{m} = (ab)^{m}

:فعّل قانون القوى

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

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

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

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

فعّل قانون فرق المربّعات

csc^{2} (x) - (cos (x) cot (x))^{2} = (csc (x) + cos (x) cot (x)) (csc (x) - cos (x) cot (x))

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

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

حلّ كل جزء على حدة

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

csc (x) + cos (x) cot (x) = 0 :

لا يوجد حلّ

csc (x) + cos (x) cot (x) = 0

sin, cos :

عبّر بواسطة

csc (x) + cos (x) cot (x)

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} + cos (x) cot (x)

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

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} + cos (x) \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} + cos (x) \frac{cos ( x )}{sin ( x )}

بسّط:

\frac{1 + cos ^{2} ( x )}{sin ( x )}

\frac{1}{sin ( x )} + cos (x) \frac{cos ( x )}{sin ( x )}

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

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

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

:اضرب كسور

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

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

cos (x) cos (x)

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

:فعّل قانون القوى

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

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

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

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

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

فعّل القانون

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

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

\frac{1 + cos ^{2} ( x )}{sin ( x )} = 0

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

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

بالاستعانة بطريقة التعويض

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

cos (x) = u :

على افتراض أنّ

1 + u^{2} = 0

1 + u^{2} = 0 : u = i, u = - i

1 + u^{2} = 0

انقل

1

إلى الجانب الأيمن

1 + u^{2} = 0

من الطرفين

1

اطرح

1 + u^{2} - 1 = 0 - 1

بسّط

u^{2} = - 1

u^{2} = - 1

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

الحلول هي

x^{2} = f (a)

لـ

u = - 1, u = - - 1

- 1

بسّط:

i

- 1

- 1 = i

:فعّل قانون الأعداد التخيليّة

= i

- - 1

بسّط:

- i

- - 1

- 1 = i

:فعّل قانون الأعداد التخيليّة

= - i

u = i, u = - i

u = cos (x)

استبدل مجددًا

cos (x) = i, cos (x) = - i

cos (x) = i, cos (x) = - i

cos (x) = i :

لا يوجد حلّ

cos (x) = i

لا يوجد حلّ

cos (x) = - i :

لا يوجد حلّ

cos (x) = - i

لا يوجد حلّ

وحّد الحلول

لا يوجد حلّ

csc (x) - cos (x) cot (x) = 0 :

لا يوجد حلّ

csc (x) - cos (x) cot (x) = 0

sin, cos :

عبّر بواسطة

csc (x) - cos (x) cot (x)

csc (x) = \frac{1}{sin ( x )}

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} - cos (x) cot (x)

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

:Use the basic trigonometric identity

= \frac{1}{sin ( x )} - cos (x) \frac{cos ( x )}{sin ( x )}

\frac{1}{sin ( x )} - cos (x) \frac{cos ( x )}{sin ( x )}

بسّط:

\frac{1 - cos ^{2} ( x )}{sin ( x )}

\frac{1}{sin ( x )} - cos (x) \frac{cos ( x )}{sin ( x )}

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

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

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

:اضرب كسور

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

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

cos (x) cos (x)

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

:فعّل قانون القوى

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

= cos^{1 + 1} (x)

1 + 1 = 2 :

اجمع الأعداد

= cos^{2} (x)

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

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

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

فعّل القانون

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

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

\frac{1 - cos ^{2} ( x )}{sin ( x )} = 0

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

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

بالاستعانة بطريقة التعويض

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

cos (x) = u :

على افتراض أنّ

1 - u^{2} = 0

1 - u^{2} = 0 : u = 1, u = - 1

1 - u^{2} = 0

انقل

1

إلى الجانب الأيمن

1 - u^{2} = 0

من الطرفين

1

اطرح

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

بسّط

- u^{2} = - 1

- u^{2} = - 1

- 1

اقسم الطرفين على

- u^{2} = - 1

- 1

اقسم الطرفين على

\frac{- u ^{2}}{- 1} = \frac{- 1}{- 1}

بسّط

u^{2} = 1

u^{2} = 1

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

الحلول هي

x^{2} = f (a)

لـ

u = 1, u = - 1

1 = 1

1

1 = 1

فعّل القانون

= 1

- 1 = - 1

- 1

1 = 1

فعّل القانون

= - 1

u = 1, u = - 1

u = cos (x)

استبدل مجددًا

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

cos (x) = 1 :

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 = 0 + 2 πn

x = 0 + 2 πn

x = 0 + 2 πn

حلّ:

x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

cos (x) = - 1 :

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 = π + 2 πn

x = π + 2 πn

وحّد الحلول

x = 2 πn, x = π + 2 πn

2 πn, π + 2 πn :

بما أنّ المعادلة غير معرّفة لـ

لا يوجد حلّ

وحّد الحلول

x \in R لا يوجد حلّ لـ

رسم

أمثلة شائعة

sin(2p+1)=-1

sin (2 p + 1) = - 1

solvefor y,a*z=5sin(2y)

so l v e f or y, a \cdot z = 5 sin (2 y)

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sin^{3} (x) = sin^{2} (x)

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

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

3cos^2(x)+4cos(x)+1=0

3 cos^{2} (x) + 4 cos (x) + 1 = 0