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شائع علم المثلثات >

(cos(x)-cos^2(x))/(sin(x))=sin(x)cos(x)

الحلّ

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

الحلّ

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

درجات

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

sin (x) cos (x)

اطرح

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

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

بسّط:

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

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

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

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

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

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

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

sin (x) cos (x) sin (x)

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

:فعّل قانون القوى

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

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

1 + 1 = 2 :

اجمع الأعداد

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

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

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

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

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

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

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

حلل إلى عوامل:

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

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

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

:فعّل قانون القوى

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

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

cos (x)

قم باخراج العامل المشترك

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

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

حلّ كل جزء على حدة

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

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

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

Rewrite using trig identities

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

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

:فعّل نطريّة فيتاغوروس

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

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

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

بالاستعانة بطريقة التعويض

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

cos (x) = u :

على افتراض أنّ

- u + u^{2} = 0

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

- u + u^{2} = 0

a x^{2} + b x + c = 0

اكتب بالصورة الاعتياديّة

u^{2} - u = 0

حلّ بالاستعانة بالصيغة التربيعيّة

u^{2} - u = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 1, b = - 1, c = 0

لـ

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

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

(- 1)^{2} - 4 \cdot 1 \cdot 0 = 1

(- 1)^{2} - 4 \cdot 1 \cdot 0

(- 1)^{2} = 1

(- 1)^{2}

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

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

= 1^{2}

1^{a} = 1

فعّل القانون

= 1

4 \cdot 1 \cdot 0 = 0

4 \cdot 1 \cdot 0

0 \cdot a = 0

فعّل القانون

= 0

= 1 - 0

1 - 0 = 1 :

اطرح الأعداد

= 1

1 = 1

فعّل القانون

= 1

u_{1, 2} = \frac{- ( - 1 ) \pm 1}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 1}{2 \cdot 1}, u_{2} = \frac{- ( - 1 ) - 1}{2 \cdot 1}

u = \frac{- ( - 1 ) + 1}{2 \cdot 1} : 1

\frac{- ( - 1 ) + 1}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{1 + 1}{2 \cdot 1}

1 + 1 = 2 :

اجمع الأعداد

= \frac{2}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{2}{2}

\frac{a}{a} = 1

فعّل القانون

= 1

u = \frac{- ( - 1 ) - 1}{2 \cdot 1} : 0

\frac{- ( - 1 ) - 1}{2 \cdot 1}

- (- a) = a

فعّل القانون

= \frac{1 - 1}{2 \cdot 1}

1 - 1 = 0 :

اطرح الأعداد

= \frac{0}{2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{0}{2}

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

فعّل القانون

= 0

حلول المعادلة التربيعيّة هي

u = 1, u = 0

u = cos (x)

استبدل مجددًا

cos (x) = 1, cos (x) = 0

cos (x) = 1, cos (x) = 0

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

وحّد الحلول

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

2 πn :

بما أنّ المعادلة غير معرّفة لـ

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

رسم

أمثلة شائعة

1=-cot^2(x)+2

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

60*pi/(180)=2arctan(x/(17))

60 \cdot \frac{π}{180} = 2 arctan (\frac{x}{17})

sin(a)=-1/8 ,(3pi)/2 <= a<= 2pi

sin (a) = - \frac{1}{8}, \frac{3 π}{2} \leq a \leq 2 π

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

3 cos^{2} (x) - 23 cos (x) = sin^{2} (x)

sin(x)+cos(x)= 3/2

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