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شائع علم المثلثات >

cot(x)>cot(1/x)

الحلّ

cot (x) > cot (\frac{1}{x})

الحلّ

2 πn < x < - \frac{9 π - 81 π ^{2} + 4}{2} + 2 πn or \frac{1}{9 π} + 2 πn < x < - 4 π + 16 π^{2} + 1 + 2 πn or \frac{1}{8 π} + 2 πn < x < - \frac{7 π - 49 π ^{2} + 4}{2} + 2 πn or \frac{1}{7 π} + 2 πn < x < - 3 π + 9 π^{2} + 1 + 2 πn or \frac{1}{6 π} + 2 πn < x < - \frac{5 π - 25 π ^{2} + 4}{2} + 2 πn or \frac{1}{5 π} + 2 πn < x < - 2 π + 4 π^{2} + 1 + 2 πn or \frac{1}{4 π} + 2 πn < x < - \frac{3 π - 9 π ^{2} + 4}{2} + 2 πn or \frac{1}{3 π} + 2 πn < x < - π + π^{2} + 1 + 2 πn or \frac{1}{2 π} + 2 πn < x < - \frac{π - π ^{2} + 4}{2} + 2 πn or \frac{1}{π} + 2 πn < x < 1 + 2 πn or π + 2 πn < x < - \frac{- π - π ^{2} + 4}{2} + 2 πn

تدوين الفاصل الزمني

(2 πn, - \frac{9 π - 81 π ^{2} + 4}{2} + 2 πn) \cup (\frac{1}{9 π} + 2 πn, - 4 π + 16 π^{2} + 1 + 2 πn) \cup (\frac{1}{8 π} + 2 πn, - \frac{7 π - 49 π ^{2} + 4}{2} + 2 πn) \cup (\frac{1}{7 π} + 2 πn, - 3 π + 9 π^{2} + 1 + 2 πn) \cup (\frac{1}{6 π} + 2 πn, - \frac{5 π - 25 π ^{2} + 4}{2} + 2 πn) \cup (\frac{1}{5 π} + 2 πn, - 2 π + 4 π^{2} + 1 + 2 πn) \cup (\frac{1}{4 π} + 2 πn, - \frac{3 π - 9 π ^{2} + 4}{2} + 2 πn) \cup (\frac{1}{3 π} + 2 πn, - π + π^{2} + 1 + 2 πn) \cup (\frac{1}{2 π} + 2 πn, - \frac{π - π ^{2} + 4}{2} + 2 πn) \cup (\frac{1}{π} + 2 πn, 1 + 2 πn) \cup (π + 2 πn, - \frac{- π - π ^{2} + 4}{2} + 2 πn)

عشري

2 πn < x < 0.03532 \dots + 2 πn or 0.03536 \dots + 2 πn < x < 0.03972 \dots + 2 πn or 0.03978 \dots + 2 πn < x < 0.04537 \dots + 2 πn or 0.04547 \dots + 2 πn < x < 0.05290 \dots + 2 πn or 0.05305 \dots + 2 πn < x < 0.06340 \dots + 2 πn or 0.06366 \dots + 2 πn < x < 0.07907 \dots + 2 πn or 0.07957 \dots + 2 πn < x < 0.10493 \dots + 2 πn or 0.10610 \dots + 2 πn < x < 0.15531 \dots + 2 πn or 0.15915 \dots + 2 πn < x < 0.29129 \dots + 2 πn or 0.31830 \dots + 2 πn < x < 1 + 2 πn or 3.14159 \dots + 2 πn < x < 3.43289 \dots + 2 πn

خطوات الحلّ

cot (x) > cot (\frac{1}{x})

انقل

cot (\frac{1}{x})

إلى الجانب الأيسر

cot (x) > cot (\frac{1}{x})

من الطرفين

cot (\frac{1}{x})

اطرح

cot (x) - cot (\frac{1}{x}) > cot (\frac{1}{x}) - cot (\frac{1}{x})

cot (x) - cot (\frac{1}{x}) > 0

cot (x) - cot (\frac{1}{x}) > 0

cot (x) - cot (\frac{1}{x})

دوريّة:

Not periodic

غير دوريّة

cot (x) - cot (\frac{1}{x})

الدالّة

= Not periodic

sin, cos :

عبّر بواسطة

cot (x) - cot (\frac{1}{x}) > 0

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

:Use the basic trigonometric identity

\frac{cos ( x )}{sin ( x )} - cot (\frac{1}{x}) > 0

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

:Use the basic trigonometric identity

\frac{cos ( x )}{sin ( x )} - \frac{cos ( \frac{1}{x} )}{sin ( \frac{1}{x} )} > 0

\frac{cos ( x )}{sin ( x )} - \frac{cos ( \frac{1}{x} )}{sin ( \frac{1}{x} )} > 0

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

بسّط:

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

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

sin (x), sin (\frac{1}{x})

المضاعف المشترك الأصغر لـ:

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

sin (x), sin (\frac{1}{x})

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (x)

sin (\frac{1}{x})

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

اكتب مجددًا الكسور بحيث يكون المقام مشترك

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

اضرب كل بسط ومقام بتعبير الذي يؤدّي إلى مقام مشترك

For

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

multiply the denominator and numerator by

sin (\frac{1}{x})

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

For

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

multiply the denominator and numerator by

sin (x)

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

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

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

:بما أنّ المقامات متساوية، اجمع البسوط

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

\frac{cos ( x ) sin ( \frac{1}{x} ) - cos ( \frac{1}{x} ) sin ( x )}{sin ( x ) sin ( \frac{1}{x} )} > 0

Find the zeroes and undifined points of

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

for

0 \leq x < 2 π

To find the zeroes, set the inequality to zero

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

\frac{cos ( x ) sin ( \frac{1}{x} ) - cos ( \frac{1}{x} ) sin ( x )}{sin ( x ) sin ( \frac{1}{x} )} = 0, 0 \leq x < 2 π : x = 1, x = - \frac{π - π ^{2} + 4}{2}, x = - π + π^{2} + 1, x = - \frac{3 π - 9 π ^{2} + 4}{2}, x = - 2 π + 4 π^{2} + 1, x = - \frac{5 π - 25 π ^{2} + 4}{2}, x = - 3 π + 9 π^{2} + 1, x = - \frac{7 π - 49 π ^{2} + 4}{2}, x = - 4 π + 16 π^{2} + 1, x = - \frac{9 π - 81 π ^{2} + 4}{2}, x = - \frac{- π - π ^{2} + 4}{2}

\frac{cos ( x ) sin ( \frac{1}{x} ) - cos ( \frac{1}{x} ) sin ( x )}{sin ( x ) sin ( \frac{1}{x} )} = 0, 0 \leq x < 2 π

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

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

Rewrite using trig identities

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

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

:فعّل متطابقة الطرح لزوايا

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

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

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

\frac{1}{x} - x = 0 + 2 πn, \frac{1}{x} - x = π + 2 πn

\frac{1}{x} - x = 0 + 2 πn, \frac{1}{x} - x = π + 2 πn

\frac{1}{x} - x = 0 + 2 πn

حلّ:

x = - πn - π^{2} n^{2} + 1, x = - πn + π^{2} n^{2} + 1

\frac{1}{x} - x = 0 + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} - x = 0 + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} x - xx = 0 \cdot x + 2 πn x

بسّط

\frac{1}{x} x - xx = 0 \cdot x + 2 πn x

\frac{1}{x} x

بسّط:

1

\frac{1}{x} x

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

:اضرب كسور

= \frac{1 \cdot x}{x}

x :

إلغ العوامل المشتركة

= 1

- xx

بسّط:

- x^{2}

- xx

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

:فعّل قانون القوى

xx = x^{1 + 1}

= - x^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= - x^{2}

0 \cdot x

بسّط:

0

0 \cdot x

0 \cdot a = 0

فعّل القانون

= 0

1 - x^{2} = 0 + 2 πn x

0 + 2 πn x

بسّط:

2 πn x

0 + 2 πn x

0 + 2 πn x = 2 πn x

= 2 πn x

1 - x^{2} = 2 πn x

1 - x^{2} = 2 πn x

1 - x^{2} = 2 πn x

1 - x^{2} = 2 πn x

حلّ:

x = - πn - π^{2} n^{2} + 1, x = - πn + π^{2} n^{2} + 1

1 - x^{2} = 2 πn x

انقل

2 πn x

إلى الجانب الأيسر

1 - x^{2} = 2 πn x

من الطرفين

2 πn x

اطرح

1 - x^{2} - 2 πn x = 2 πn x - 2 πn x

بسّط

1 - x^{2} - 2 πn x = 0

1 - x^{2} - 2 πn x = 0

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

اكتب بالصورة الاعتياديّة

- x^{2} - 2 πn x + 1 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- x^{2} - 2 πn x + 1 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 1, b = - 2 πn, c = 1

لـ

x_{1, 2} = \frac{- ( - 2 πn ) \pm ( - 2 πn ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

x_{1, 2} = \frac{- ( - 2 πn ) \pm ( - 2 πn ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 2 πn)^{2} - 4 (- 1) \cdot 1

بسّط:

2 π^{2} n^{2} + 1

(- 2 πn)^{2} - 4 (- 1) \cdot 1

- (- a) = a

فعّل القانون

= (- 2 πn)^{2} + 4 \cdot 1 \cdot 1

(- 2 πn)^{2} = 2^{2} π^{2} n^{2}

(- 2 πn)^{2}

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

(- 2 πn)^{2} = (2 πn)^{2}

= (2 πn)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 2^{2} π^{2} n^{2}

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

4 \cdot 1 \cdot 1 = 4 :

اضرب الأعداد

= 4

= 2^{2} π^{2} n^{2} + 4

2^{2} π^{2} n^{2} + 4

حلل إلى عوامل:

4 (π^{2} n^{2} + 1)

2^{2} π^{2} n^{2} + 4

أعد الكتابة كـ

= 4 π^{2} n^{2} + 4 \cdot 1

4

قم باخراج العامل المشترك

= 4 (π^{2} n^{2} + 1)

= 4 (π^{2} n^{2} + 1)

a \geq 0, b \geq 0

بافتراض أنّ

n ab = n a n b

:فعّل قانون الجذور

= 4 π^{2} n^{2} + 1

4 = 2

4

4 = 2^{2} :

حلّل العدد لعوامله أوّليّة

= 2^{2}

n a^{n} = a

:فعْل قانون الجذور

2^{2} = 2

= 2

= 2 π^{2} n^{2} + 1

x_{1, 2} = \frac{- ( - 2 πn ) \pm 2 π ^{2} n ^{2} + 1}{2 ( - 1 )}

Separate the solutions

x_{1} = \frac{- ( - 2 πn ) + 2 π ^{2} n ^{2} + 1}{2 ( - 1 )}, x_{2} = \frac{- ( - 2 πn ) - 2 π ^{2} n ^{2} + 1}{2 ( - 1 )}

x = \frac{- ( - 2 πn ) + 2 π ^{2} n ^{2} + 1}{2 ( - 1 )} : - πn - π^{2} n^{2} + 1

\frac{- ( - 2 πn ) + 2 π ^{2} n ^{2} + 1}{2 ( - 1 )}

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

:احذف الأقواس

= \frac{2 πn + 2 π ^{2} n ^{2} + 1}{- 2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{2 πn + 2 π ^{2} n ^{2} + 1}{- 2}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{2 πn + 2 π ^{2} n ^{2} + 1}{2}

\frac{2 πn + 2 π ^{2} n ^{2} + 1}{2}

اختزل:

πn + π^{2} n^{2} + 1

\frac{2 πn + 2 π ^{2} n ^{2} + 1}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( πn + 1 + n ^{2} π ^{2} )}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn + π^{2} n^{2} + 1

= - (πn + π^{2} n^{2} + 1)

افتح أقواس

= - (πn) - (π^{2} n^{2} + 1)

فعّل قوانين سالب-موجب

+ (- a) = - a

= - πn - π^{2} n^{2} + 1

x = \frac{- ( - 2 πn ) - 2 π ^{2} n ^{2} + 1}{2 ( - 1 )} : - πn + π^{2} n^{2} + 1

\frac{- ( - 2 πn ) - 2 π ^{2} n ^{2} + 1}{2 ( - 1 )}

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

:احذف الأقواس

= \frac{2 πn - 2 π ^{2} n ^{2} + 1}{- 2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{2 πn - 2 π ^{2} n ^{2} + 1}{- 2}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{2 πn - 2 π ^{2} n ^{2} + 1}{2}

\frac{2 πn - 2 π ^{2} n ^{2} + 1}{2}

اختزل:

πn - π^{2} n^{2} + 1

\frac{2 πn - 2 π ^{2} n ^{2} + 1}{2}

2

قم باخراج العامل المشترك

= \frac{2 ( πn - 1 + n ^{2} π ^{2} )}{2}

\frac{2}{2} = 1 :

اقسم الأعداد

= πn - π^{2} n^{2} + 1

= - (πn - π^{2} n^{2} + 1)

افتح أقواس

= - (πn) - (- π^{2} n^{2} + 1)

فعّل قوانين سالب-موجب

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

= - πn + π^{2} n^{2} + 1

حلول المعادلة التربيعيّة هي

x = - πn - π^{2} n^{2} + 1, x = - πn + π^{2} n^{2} + 1

x = - πn - π^{2} n^{2} + 1, x = - πn + π^{2} n^{2} + 1

\frac{1}{x} - x = π + 2 πn

حلّ:

x = - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}, x = - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

\frac{1}{x} - x = π + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} - x = π + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} x - xx = π x + 2 πn x

بسّط

\frac{1}{x} x - xx = π x + 2 πn x

\frac{1}{x} x

بسّط:

1

\frac{1}{x} x

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

:اضرب كسور

= \frac{1 \cdot x}{x}

x :

إلغ العوامل المشتركة

= 1

- xx

بسّط:

- x^{2}

- xx

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

:فعّل قانون القوى

xx = x^{1 + 1}

= - x^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= - x^{2}

1 - x^{2} = π x + 2 πn x

1 - x^{2} = π x + 2 πn x

1 - x^{2} = π x + 2 πn x

1 - x^{2} = π x + 2 πn x

حلّ:

x = - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}, x = - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

1 - x^{2} = π x + 2 πn x

انقل

2 πn x

إلى الجانب الأيسر

1 - x^{2} = π x + 2 πn x

من الطرفين

2 πn x

اطرح

1 - x^{2} - 2 πn x = π x + 2 πn x - 2 πn x

بسّط

1 - x^{2} - 2 πn x = π x

1 - x^{2} - 2 πn x = π x

انقل

π x

إلى الجانب الأيسر

1 - x^{2} - 2 πn x = π x

من الطرفين

π x

اطرح

1 - x^{2} - 2 πn x - π x = π x - π x

بسّط

1 - x^{2} - 2 πn x - π x = 0

1 - x^{2} - 2 πn x - π x = 0

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

اكتب بالصورة الاعتياديّة

- x^{2} - (2 πn + π) x + 1 = 0

حلّ بالاستعانة بالصيغة التربيعيّة

- x^{2} - (2 πn + π) x + 1 = 0

الصيغة لحلّ المعادلة التربيعيّة

a = - 1, b = - 2 πn - π, c = 1

لـ

x_{1, 2} = \frac{- ( - 2 πn - π ) \pm ( - 2 πn - π ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

x_{1, 2} = \frac{- ( - 2 πn - π ) \pm ( - 2 πn - π ) ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

(- 2 πn - π)^{2} - 4 (- 1) \cdot 1

بسّط:

(- 2 πn - π)^{2} + 4

(- 2 πn - π)^{2} - 4 (- 1) \cdot 1

- (- a) = a

فعّل القانون

= (- 2 πn - π)^{2} + 4 \cdot 1 \cdot 1

4 \cdot 1 \cdot 1 = 4 :

اضرب الأعداد

= (- 2 πn - π)^{2} + 4

x_{1, 2} = \frac{- ( - 2 πn - π ) \pm ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )}

Separate the solutions

x_{1} = \frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )}, x_{2} = \frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )}

x = \frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )} : - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}

\frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )}

(- a) = - a

:احذف الأقواس

= \frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{- 2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{- 2}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{- ( - 2 πn - π ) + ( - 2 πn - π ) ^{2} + 4}{2}

- (- 2 πn - π) : 2 πn + π

- (- 2 πn - π)

افتح أقواس

= - (- 2 πn) - (- π)

فعّل قوانين سالب-موجب

- (- a) = a

= 2 πn + π

= - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}

x = \frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )} : - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

\frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{2 ( - 1 )}

(- a) = - a

:احذف الأقواس

= \frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{- 2 \cdot 1}

2 \cdot 1 = 2 :

اضرب الأعداد

= \frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{- 2}

\frac{a}{- b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{- ( - 2 πn - π ) - ( - 2 πn - π ) ^{2} + 4}{2}

- (- 2 πn - π) : 2 πn + π

- (- 2 πn - π)

افتح أقواس

= - (- 2 πn) - (- π)

فعّل قوانين سالب-موجب

- (- a) = a

= 2 πn + π

= - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

حلول المعادلة التربيعيّة هي

x = - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}, x = - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

x = - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}, x = - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

x = - πn - π^{2} n^{2} + 1, x = - πn + π^{2} n^{2} + 1, x = - \frac{2 πn + π + ( - 2 πn - π ) ^{2} + 4}{2}, x = - \frac{2 πn + π - ( - 2 πn - π ) ^{2} + 4}{2}

0 \leq x < 2 π :

حلول للمدى

x = 1, x = - \frac{π - π ^{2} + 4}{2}, x = - π + π^{2} + 1, x = - \frac{3 π - 9 π ^{2} + 4}{2}, x = - 2 π + 4 π^{2} + 1, x = - \frac{5 π - 25 π ^{2} + 4}{2}, x = - 3 π + 9 π^{2} + 1, x = - \frac{7 π - 49 π ^{2} + 4}{2}, x = - 4 π + 16 π^{2} + 1, x = - \frac{9 π - 81 π ^{2} + 4}{2}, x = - \frac{- π - π ^{2} + 4}{2}

Find the undefined points:

x = π, x = \frac{1}{π}, x = \frac{1}{2 π}, x = \frac{1}{3 π}, x = \frac{1}{4 π}, x = \frac{1}{5 π}, x = \frac{1}{6 π}, x = \frac{1}{7 π}, x = \frac{1}{8 π}, x = \frac{1}{9 π}

Find the zeros of the denominator

sin (x) sin (\frac{1}{x}) = 0

حلّ كل جزء على حدة

sin (x) = 0 or sin (\frac{1}{x}) = 0

sin (x) = 0, 0 \leq x < 2 π : x = 0, x = π

sin (x) = 0, 0 \leq x < 2 π

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

0 \leq x < 2 π :

حلول للمدى

x = 0, x = π

sin (\frac{1}{x}) = 0, 0 \leq x < 2 π : x = \frac{1}{π}, x = \frac{1}{2 π}, x = \frac{1}{3 π}, x = \frac{1}{4 π}, x = \frac{1}{5 π}, x = \frac{1}{6 π}, x = \frac{1}{7 π}, x = \frac{1}{8 π}, x = \frac{1}{9 π}

sin (\frac{1}{x}) = 0, 0 \leq x < 2 π

sin (\frac{1}{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}

\frac{1}{x} = 0 + 2 πn, \frac{1}{x} = π + 2 πn

\frac{1}{x} = 0 + 2 πn, \frac{1}{x} = π + 2 πn

\frac{1}{x} = 0 + 2 πn

حلّ:

x = \frac{1}{2 πn}; n \neq = 0

\frac{1}{x} = 0 + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} = 0 + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} x = 0 \cdot x + 2 πn x

بسّط

1 = 0 + 2 πn x

0 + 2 πn x

بسّط:

2 πn x

0 + 2 πn x

0 + 2 πn x = 2 πn x

= 2 πn x

1 = 2 πn x

1 = 2 πn x

بدّل الأطراف

2 πn x = 1

2 πn; n \neq = 0

اقسم الطرفين على

2 πn x = 1

2 πn; n \neq = 0

اقسم الطرفين على

\frac{2 πn x}{2 πn} = \frac{1}{2 πn}; n \neq = 0

بسّط

x = \frac{1}{2 πn}; n \neq = 0

x = \frac{1}{2 πn}; n \neq = 0

\frac{1}{x} = π + 2 πn

حلّ:

x = \frac{1}{π ( 1 + 2 n )}; n \neq = - \frac{1}{2}

\frac{1}{x} = π + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} = π + 2 πn

x

اضرب الطرفين بـ

\frac{1}{x} x = π x + 2 πn x

بسّط

1 = π x + 2 πn x

1 = π x + 2 πn x

بدّل الأطراف

π x + 2 πn x = 1

π x + 2 πn x

حلل إلى عوامل:

π x (1 + 2 n)

π x + 2 πn x

x π

قم باخراج العامل المشترك

= x π (1 + 2 n)

π x (1 + 2 n) = 1

π (1 + 2 n); n \neq = - \frac{1}{2}

اقسم الطرفين على

π x (1 + 2 n) = 1

π (1 + 2 n); n \neq = - \frac{1}{2}

اقسم الطرفين على

\frac{π x ( 1 + 2 n )}{π ( 1 + 2 n )} = \frac{1}{π ( 1 + 2 n )}; n \neq = - \frac{1}{2}

بسّط

x = \frac{1}{π ( 1 + 2 n )}; n \neq = - \frac{1}{2}

x = \frac{1}{π ( 1 + 2 n )}; n \neq = - \frac{1}{2}

x = \frac{1}{2 πn}, x = \frac{1}{π ( 1 + 2 n )}; n \neq = 0, n \neq = - \frac{1}{2}

0 \leq x < 2 π :

حلول للمدى

x = \frac{1}{π}, x = \frac{1}{2 π}, x = \frac{1}{3 π}, x = \frac{1}{4 π}, x = \frac{1}{5 π}, x = \frac{1}{6 π}, x = \frac{1}{7 π}, x = \frac{1}{8 π}, x = \frac{1}{9 π}

وحّد الحلول

x = 0, x = π, x = \frac{1}{π}, x = \frac{1}{2 π}, x = \frac{1}{3 π}, x = \frac{1}{4 π}, x = \frac{1}{5 π}, x = \frac{1}{6 π}, x = \frac{1}{7 π}, x = \frac{1}{8 π}, x = \frac{1}{9 π}

0 :

بما أنّ المعادلة غير معرّفة لـ

x = π, x = \frac{1}{π}, x = \frac{1}{2 π}, x = \frac{1}{3 π}, x = \frac{1}{4 π}, x = \frac{1}{5 π}, x = \frac{1}{6 π}, x = \frac{1}{7 π}, x = \frac{1}{8 π}, x = \frac{1}{9 π}

- \frac{9 π - 81 π ^{2} + 4}{2}, \frac{1}{9 π}, - 4 π + 16 π^{2} + 1, \frac{1}{8 π}, - \frac{7 π - 49 π ^{2} + 4}{2}, \frac{1}{7 π}, - 3 π + 9 π^{2} + 1, \frac{1}{6 π}, - \frac{5 π - 25 π ^{2} + 4}{2}, \frac{1}{5 π}, - 2 π + 4 π^{2} + 1, \frac{1}{4 π}, - \frac{3 π - 9 π ^{2} + 4}{2}, \frac{1}{3 π}, - π + π^{2} + 1, \frac{1}{2 π}, - \frac{π - π ^{2} + 4}{2}, \frac{1}{π}, 1, π, - \frac{- π - π ^{2} + 4}{2}

ميّز المقاطع المختلفة

0 < x < - \frac{9 π - 81 π ^{2} + 4}{2}, - \frac{9 π - 81 π ^{2} + 4}{2} < x < \frac{1}{9 π}, \frac{1}{9 π} < x < - 4 π + 16 π^{2} + 1, - 4 π + 16 π^{2} + 1 < x < \frac{1}{8 π}, \frac{1}{8 π} < x < - \frac{7 π - 49 π ^{2} + 4}{2}, - \frac{7 π - 49 π ^{2} + 4}{2} < x < \frac{1}{7 π}, \frac{1}{7 π} < x < - 3 π + 9 π^{2} + 1, - 3 π + 9 π^{2} + 1 < x < \frac{1}{6 π}, \frac{1}{6 π} < x < - \frac{5 π - 25 π ^{2} + 4}{2}, - \frac{5 π - 25 π ^{2} + 4}{2} < x < \frac{1}{5 π}, \frac{1}{5 π} < x < - 2 π + 4 π^{2} + 1, - 2 π + 4 π^{2} + 1 < x < \frac{1}{4 π}, \frac{1}{4 π} < x < - \frac{3 π - 9 π ^{2} + 4}{2}, - \frac{3 π - 9 π ^{2} + 4}{2} < x < \frac{1}{3 π}, \frac{1}{3 π} < x < - π + π^{2} + 1, - π + π^{2} + 1 < x < \frac{1}{2 π}, \frac{1}{2 π} < x < - \frac{π - π ^{2} + 4}{2}, - \frac{π - π ^{2} + 4}{2} < x < \frac{1}{π}, \frac{1}{π} < x < 1, 1 < x < π, π < x < - \frac{- π - π ^{2} + 4}{2}, - \frac{- π - π ^{2} + 4}{2} < x < 2 π

لخّص في جدول

> 0 :

ميّز المقاطع التي تحقّق الشرط

0 < x < - \frac{9 π - 81 π ^{2} + 4}{2} or \frac{1}{9 π} < x < - 4 π + 16 π^{2} + 1 or \frac{1}{8 π} < x < - \frac{7 π - 49 π ^{2} + 4}{2} or \frac{1}{7 π} < x < - 3 π + 9 π^{2} + 1 or \frac{1}{6 π} < x < - \frac{5 π - 25 π ^{2} + 4}{2} or \frac{1}{5 π} < x < - 2 π + 4 π^{2} + 1 or \frac{1}{4 π} < x < - \frac{3 π - 9 π ^{2} + 4}{2} or \frac{1}{3 π} < x < - π + π^{2} + 1 or \frac{1}{2 π} < x < - \frac{π - π ^{2} + 4}{2} or \frac{1}{π} < x < 1 or π < x < - \frac{- π - π ^{2} + 4}{2}

cot (x) - cot (\frac{1}{x}) :

استخدم دوريّة الـ

2 πn < x < - \frac{9 π - 81 π ^{2} + 4}{2} + 2 πn or \frac{1}{9 π} + 2 πn < x < - 4 π + 16 π^{2} + 1 + 2 πn or \frac{1}{8 π} + 2 πn < x < - \frac{7 π - 49 π ^{2} + 4}{2} + 2 πn or \frac{1}{7 π} + 2 πn < x < - 3 π + 9 π^{2} + 1 + 2 πn or \frac{1}{6 π} + 2 πn < x < - \frac{5 π - 25 π ^{2} + 4}{2} + 2 πn or \frac{1}{5 π} + 2 πn < x < - 2 π + 4 π^{2} + 1 + 2 πn or \frac{1}{4 π} + 2 πn < x < - \frac{3 π - 9 π ^{2} + 4}{2} + 2 πn or \frac{1}{3 π} + 2 πn < x < - π + π^{2} + 1 + 2 πn or \frac{1}{2 π} + 2 πn < x < - \frac{π - π ^{2} + 4}{2} + 2 πn or \frac{1}{π} + 2 πn < x < 1 + 2 πn or π + 2 πn < x < - \frac{- π - π ^{2} + 4}{2} + 2 πn

أمثلة شائعة

(2sin(x)+sqrt(2))/(cos(x))<= 0

\frac{2 sin ( x ) + 2}{cos ( x )} \leq 0

1/2 >cos(x)

\frac{1}{2} > cos (x)

2cos^2(a)-1>=-1/8

2 cos^{2} (a) - 1 \geq - \frac{1}{8}

4sin^2(x)-3<0,-2pi<= x<= 0

4 sin^{2} (x) - 3 < 0, - 2 π \leq x \leq 0

sin^2(x)>0.5

sin^{2} (x) > 0.5