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شائع علم المثلثات >

tan^2(x)-sin(x)=tan^2(x)sin^2(x)

الحلّ

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

الحلّ

x = 2 πn, x = π + 2 πn

درجات

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

خطوات الحلّ

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

من الطرفين

tan^{2} (x) sin^{2} (x)

اطرح

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

sin, cos :

عبّر بواسطة

- sin (x) + tan^{2} (x) - sin^{2} (x) tan^{2} (x)

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

:Use the basic trigonometric identity

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

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

بسّط:

\frac{- cos ^{2} ( x ) sin ( x ) + sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

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

:فعّل قانون القوى

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

sin^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

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

:فعّل قانون القوى

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

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

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

:اضرب كسور

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

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

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

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

:فعّل قانون القوى

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

= sin^{2 + 2} (x)

2 + 2 = 4 :

اجمع الأعداد

= sin^{4} (x)

= \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin (x) + \frac{sin ^{2} ( x )}{cos ^{2} ( x )} - \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

\frac{sin ^{2} ( x )}{cos ^{2} ( x )} - \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

وحّد الكسور:

\frac{sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

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

فعّل القانون

= \frac{sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin (x) + \frac{sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

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

:حوّل الأعداد لكسور

= - \frac{sin ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- sin ( x ) cos ^{2} ( x ) + sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) sin ( x ) + sin ^{2} ( x ) - sin ^{4} ( x )}{cos ^{2} ( x )}

\frac{sin ^{2} ( x ) - sin ^{4} ( x ) - cos ^{2} ( x ) sin ( x )}{cos ^{2} ( x )} = 0

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

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

sin^{2} (x) - sin^{4} (x) - cos^{2} (x) sin (x)

حلل إلى عوامل:

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

sin^{2} (x) - sin^{4} (x) - cos^{2} (x) sin (x)

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

:فعّل قانون القوى

sin^{4} (x) = sin (x) sin^{3} (x), sin^{2} (x) = sin (x) sin (x)

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

sin (x)

قم باخراج العامل المشترك

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

sin (x) (sin (x) - sin^{3} (x) - cos^{2} (x)) = 0

حلّ كل جزء على حدة

sin (x) = 0 or sin (x) - sin^{3} (x) - cos^{2} (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

sin (x) = 0 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

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

sin (x) - sin^{3} (x) - cos^{2} (x) = 0

Rewrite using trig identities

- cos^{2} (x) + sin (x) - sin^{3} (x)

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

:فعّل نطريّة فيتاغوروس

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

= - (1 - sin^{2} (x)) + sin (x) - sin^{3} (x)

- (1 - sin^{2} (x)) : - 1 + sin^{2} (x)

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

افتح أقواس

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

فعّل قوانين سالب-موجب

- (- a) = a, - (a) = - a

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

= - 1 + sin^{2} (x) + sin (x) - sin^{3} (x)

- 1 + sin (x) + sin^{2} (x) - sin^{3} (x) = 0

بالاستعانة بطريقة التعويض

- 1 + sin (x) + sin^{2} (x) - sin^{3} (x) = 0

sin (x) = u :

على افتراض أنّ

- 1 + u + u^{2} - u^{3} = 0

- 1 + u + u^{2} - u^{3} = 0 : u = 1, u = - 1

- 1 + u + u^{2} - u^{3} = 0

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

اكتب بالصورة الاعتياديّة

- u^{3} + u^{2} + u - 1 = 0

- u^{3} + u^{2} + u - 1

حلّل إلى عوامل:

- (u - 1)^{2} (u + 1)

- u^{3} + u^{2} + u - 1

- 1

قم باخراج العامل المشترك

= - (u^{3} - u^{2} - u + 1)

u^{3} - u^{2} - u + 1

حلل إلى عوامل:

(u - 1) (u + 1) (u - 1)

u^{3} - u^{2} - u + 1

= (u^{3} - u^{2}) + (- u + 1)

- (u - 1)

- u + 1

من

- 1

اخرج العامل

- u + 1

- 1

قم باخراج العامل المشترك

= - (u - 1)

u^{2} (u - 1)

u^{3} - u^{2}

من

u^{2}

اخرج العامل

u^{3} - u^{2}

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

:فعّل قانون القوى

u^{3} = u u^{2}

= u u^{2} - u^{2}

u^{2}

قم باخراج العامل المشترك

= u^{2} (u - 1)

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

u - 1

قم باخراج العامل المشترك

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

u^{2} - 1

حلل إلى عوامل:

(u + 1) (u - 1)

u^{2} - 1

1^{2}

كـ

1

اكتب مجددًا

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

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

فعّل قانون فرق المربّعات

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

= (u + 1) (u - 1)

= (u - 1) (u + 1) (u - 1)

= - (u - 1) (u + 1) (u - 1)

بسّط

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

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

حلّ عن طريق مساواة العوامل لصفر

u - 1 = 0 or u + 1 = 0

u - 1 = 0

حلّ:

u = 1

u - 1 = 0

انقل

1

إلى الجانب الأيمن

u - 1 = 0

للطرفين

1

أضف

u - 1 + 1 = 0 + 1

بسّط

u = 1

u = 1

u + 1 = 0

حلّ:

u = - 1

u + 1 = 0

انقل

1

إلى الجانب الأيمن

u + 1 = 0

من الطرفين

1

اطرح

u + 1 - 1 = 0 - 1

بسّط

u = - 1

u = - 1

The solutions are

u = 1, u = - 1

u = sin (x)

استبدل مجددًا

sin (x) = 1, sin (x) = - 1

sin (x) = 1, sin (x) = - 1

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

sin (x) = 1 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

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

sin (x) = - 1

sin (x) = - 1 :

حلول عامّة لـ

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 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

وحّد الحلول

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

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

بما أنّ المعادلة غير معرّفة لـ

x = 2 πn, x = π + 2 πn

رسم

أمثلة شائعة

3cos(a)-1=0

3 cos (a) - 1 = 0

tan^2(x)+tan(x)+cot(x)+cot^2(x)=4

tan^{2} (x) + tan (x) + cot (x) + cot^{2} (x) = 4

tan(a)=0

tan (a) = 0

cos^2(x)+3|cos(x)|-1=0

cos^{2} (x) + 3 ∣ cos (x) ∣ - 1 = 0

cos^5(x)=sin(75)

cos^{5} (x) = sin (7 5^{\circ})