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شائع علم المثلثات >

1/(cos(x))+sec^2(x)-1-cos(x)=0

الحلّ

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

الحلّ

x = π + 2 πn, x = 2 πn

درجات

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

خطوات الحلّ

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

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

بسّط:

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

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

sec^{2} (x) = \frac{sec ^{2} ( x ) cos ( x )}{cos ( x )}, 1 = \frac{1 cos ( x )}{cos ( x )}, cos (x) = \frac{cos ( x ) cos ( x )}{cos ( x )}

:حوّل الأعداد لكسور

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

1 + sec^{2} (x) cos (x) - 1 \cdot cos (x) - cos (x) cos (x) = 1 + sec^{2} (x) cos (x) - cos (x) - cos^{2} (x)

1 + sec^{2} (x) cos (x) - 1 \cdot cos (x) - cos (x) cos (x)

1 \cdot cos (x) = cos (x)

1 \cdot cos (x)

1 \cdot cos (x) = cos (x) :

اضرب

= cos (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)

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

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

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

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

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

Rewrite using trig identities

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

cos (x) = \frac{1}{sec ( x )}

:Use the basic trigonometric identity

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

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

بسّط:

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

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

(\frac{1}{sec ( x )})^{2} = \frac{1}{sec ^{2} ( x )}

(\frac{1}{sec ( x )})^{2}

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

:فعّل قانون القوى

= \frac{1 ^{2}}{sec ^{2} ( x )}

1^{a} = 1

فعّل القانون

1^{2} = 1

= \frac{1}{sec ^{2} ( x )}

\frac{1}{sec ( x )} sec^{2} (x) = sec (x)

\frac{1}{sec ( x )} sec^{2} (x)

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

:اضرب كسور

= \frac{1 \cdot sec ^{2} ( x )}{sec ( x )}

1 \cdot sec^{2} (x) = sec^{2} (x) :

اضرب

= \frac{sec ^{2} ( x )}{sec ( x )}

sec (x) :

إلغ العوامل المشتركة

= sec (x)

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

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

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

بالاستعانة بطريقة التعويض

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

sec (x) = u :

على افتراض أنّ

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

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

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

اضرب بالمضاعف المشترك الأصغر

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

Find Least Common Multiplier of

u^{2}, u : u^{2}

u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2}

u

= u^{2}

u^{2} =

اضرب بالمضاعف المشترك الأصغر

1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} - \frac{1}{u} u^{2} + u u^{2} = 0 \cdot u^{2}

بسّط

1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} - \frac{1}{u} u^{2} + u u^{2} = 0 \cdot u^{2}

1 \cdot u^{2}

بسّط:

u^{2}

1 \cdot u^{2}

1 \cdot u^{2} = u^{2} :

اضرب

= u^{2}

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

بسّط:

- 1

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

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

:اضرب كسور

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

u^{2} :

إلغ العوامل المشتركة

= - 1

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

بسّط:

- u

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

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

:اضرب كسور

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

1 \cdot u^{2} = u^{2} :

اضرب

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

u :

إلغ العوامل المشتركة

= - u

u u^{2}

بسّط:

u^{3}

u u^{2}

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

:فعّل قانون القوى

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

= u^{1 + 2}

1 + 2 = 3 :

اجمع الأعداد

= u^{3}

0 \cdot u^{2}

بسّط:

0

0 \cdot u^{2}

0 \cdot a = 0

فعّل القانون

= 0

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

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

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

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

حلّ:

u = - 1, u = 1

u^{2} - 1 - u + 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

= (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)

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

من الطرفين

1

اطرح

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

للطرفين

1

أضف

u - 1 + 1 = 0 + 1

بسّط

u = 1

u = 1

The solutions are

u = - 1, u = 1

u = - 1, u = 1

افحص الإجبات

جد نقاط غير معرّفة:

u = 0

وقم بمساواتها لصفر

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

خذ المقامات في

u^{2} = 0

حلّ:

u = 0

u^{2} = 0

x^{n} = 0 \Rightarrow x = 0

فعّل القانون

u = 0

u = 0

النقاط التالية غير معرّفة

u = 0

ضمّ النقاط غير المعرّفة مع الحلول

u = - 1, u = 1

u = sec (x)

استبدل مجددًا

sec (x) = - 1, sec (x) = 1

sec (x) = - 1, sec (x) = 1

sec (x) = - 1 : x = π + 2 πn

sec (x) = - 1

sec (x) = - 1 :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

sec (x) = 1 : x = 2 πn

sec (x) = 1

sec (x) = 1 :

حلول عامّة لـ

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

وحّد الحلول

x = π + 2 πn, x = 2 πn

رسم

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