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solvefor x,f=arctan(x/(sqrt(1-x^2)))

الحلّ

solve for

x, f = arctan (\frac{x}{1 - x ^{2}})

الحلّ

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

خطوات الحلّ

f = arctan (\frac{x}{1 - x ^{2}})

بدّل الأطراف

arctan (\frac{x}{1 - x ^{2}}) = f

Apply trig inverse properties

arctan (\frac{x}{1 - x ^{2}}) = f

arctan (x) = a \Rightarrow x = tan (a)

\frac{x}{1 - x ^{2}} = tan (f)

\frac{x}{1 - x ^{2}} = tan (f)

\frac{x}{1 - x ^{2}} = tan (f)

حلّ:

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{x}{1 - x ^{2}} = tan (f)

1 - x^{2}

اضرب الطرفين بـ

\frac{x}{1 - x ^{2}} 1 - x^{2} = tan (f) 1 - x^{2}

بسّط

x = tan (f) 1 - x^{2}

ربّع الطرفين:

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

x = tan (f) 1 - x^{2}

x^{2} = (tan (f) 1 - x^{2})^{2}

(tan (f) 1 - x^{2})^{2}

وسّع:

tan^{2} (f) - x^{2} tan^{2} (f)

(tan (f) 1 - x^{2})^{2}

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

:فعّل قانون القوى

= tan^{2} (f) (1 - x^{2})^{2}

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 1 - x^{2}

= tan^{2} (f) (1 - x^{2})

(1 - x^{2}) tan^{2} (f)

وسّع:

tan^{2} (f) - x^{2} tan^{2} (f)

(1 - x^{2}) tan^{2} (f)

= tan^{2} (f) (1 - x^{2})

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = tan^{2} (f), b = 1, c = x^{2}

= tan^{2} (f) \cdot 1 - tan^{2} (f) x^{2}

= 1 \cdot tan^{2} (f) - x^{2} tan^{2} (f)

1 \cdot tan^{2} (f) = tan^{2} (f) :

اضرب

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

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

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

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

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

حلّ:

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

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

انقل

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

إلى الجانب الأيسر

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

للطرفين

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

أضف

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f) - x^{2} tan^{2} (f) + x^{2} tan^{2} (f)

بسّط

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f)

x^{2} + x^{2} tan^{2} (f) = tan^{2} (f)

x^{2} + x^{2} tan^{2} (f)

حلل إلى عوامل:

x^{2} (1 + tan^{2} (f))

x^{2} + x^{2} tan^{2} (f)

x^{2}

قم باخراج العامل المشترك

= x^{2} (1 + tan^{2} (f))

x^{2} (1 + tan^{2} (f)) = tan^{2} (f)

1 + tan^{2} (f)

اقسم الطرفين على

x^{2} (1 + tan^{2} (f)) = tan^{2} (f)

1 + tan^{2} (f)

اقسم الطرفين على

\frac{x ^{2} ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

بسّط

x^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

x^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

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

الحلول هي

x^{2} = f (a)

لـ

x = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

بسّط:

\frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

a \geq 0, b \geq 0

بافتراض أنّ

:فعّل قانون الجذور

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

a \geq 0

بافتراض أنّ

:فعّل قانون الجذور

tan^{2} (f) = tan (f)

= \frac{tan ( f )}{1 + tan ^{2} ( f )}

- \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

بسّط:

- \frac{tan ( f )}{1 + tan ^{2} ( f )}

- \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

بسّط:

\frac{tan ( f )}{1 + tan ^{2} ( f )}

\frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

a \geq 0, b \geq 0

بافتراض أنّ

:فعّل قانون الجذور

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

a \geq 0

بافتراض أنّ

:فعّل قانون الجذور

tan^{2} (f) = tan (f)

= \frac{tan ( f )}{1 + tan ^{2} ( f )}

= - \frac{tan ( f )}{tan ^{2} ( f ) + 1}

= - \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

افحص الإجبات:

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

صحيح

, x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

خطأ

للتحقّق من دقّة الحلول

\frac{x}{1 - x ^{2}} = tan (f)

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

استبدل:

صحيح

\frac{( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

\frac{( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}}

بسّط:

tan (f)

\frac{\frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}}

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

: استخدم ميزات الكسور التالية

= \frac{tan ( f )}{1 + tan ^{2} ( f ) 1 - ( \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}}

1 - (\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2} = \frac{1}{1 + tan ^{2} ( f )}

1 - (\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2}

(\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2} = \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

(\frac{tan ( f )}{1 + tan ^{2} ( f )})^{2}

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

:فعّل قانون القوى

= \frac{tan ^{2} ( f )}{( 1 + tan ^{2} ( f ) ) ^{2}}

(1 + tan^{2} (f))^{2} : 1 + tan^{2} (f)

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 + tan^{2} (f))^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

= (1 + tan^{2} (f))^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 1 + tan^{2} (f)

= \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

= 1 - \frac{tan ^{2} ( f )}{tan ^{2} ( f ) + 1}

1 - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

وحّد:

\frac{1}{1 + tan ^{2} ( f )}

1 - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

1 = \frac{1 ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )}

:حوّل الأعداد لكسور

= \frac{1 \cdot ( 1 + tan ^{2} ( f ) )}{1 + tan ^{2} ( f )} - \frac{tan ^{2} ( f )}{1 + tan ^{2} ( f )}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{1 \cdot ( 1 + tan ^{2} ( f ) ) - tan ^{2} ( f )}{1 + tan ^{2} ( f )}

1 \cdot (1 + tan^{2} (f)) - tan^{2} (f) = 1

1 \cdot (1 + tan^{2} (f)) - tan^{2} (f)

1 \cdot (1 + tan^{2} (f)) = 1 + tan^{2} (f)

1 \cdot (1 + tan^{2} (f))

1 \cdot (1 + tan^{2} (f)) = (1 + tan^{2} (f)) :

اضرب

= 1 + tan^{2} (f)

(a) = a

:احذف الأقواس

= 1 + tan^{2} (f)

= 1 + tan^{2} (f) - tan^{2} (f)

tan^{2} (f) - tan^{2} (f) = 0 :

اجمع العناصر المتشابهة

= 1

= \frac{1}{1 + tan ^{2} ( f )}

= \frac{1}{1 + tan ^{2} ( f )}

a \geq 0, b \geq 0

بافتراض أنّ

:فعّل قانون الجذور

= \frac{1}{1 + tan ^{2} ( f )}

1 = 1

فعّل القانون

= \frac{1}{1 + tan ^{2} ( f )}

= \frac{tan ( f )}{\frac{1}{t a n ^{2} ( f ) + 1} tan ^{2} ( f ) + 1}

1 + tan^{2} (f) \frac{1}{1 + tan ^{2} ( f )}

اضرب بـ:

1

1 + tan^{2} (f) \frac{1}{1 + tan ^{2} ( f )}

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

:اضرب كسور

= \frac{1 \cdot 1 + tan ^{2} ( f )}{1 + tan ^{2} ( f )}

1 + tan^{2} (f) :

إلغ العوامل المشتركة

= 1

= \frac{tan ( f )}{1}

\frac{a}{1} = a

فعّل القانون

= tan (f)

tan (f) = tan (f)

صحيح

خطأ

\Leftarrow \frac{- \frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f) : x = - \frac{tan ( f )}{1 + tan ^{2} ( f )}

عوّض

\frac{( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} )}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

بالاستعانة بطريقة التعويض

\frac{- \frac{t a n ( f )}{1 + t a n ^{2} ( f )}}{1 - ( - \frac{t a n ( f )}{1 + t a n ^{2} ( f )} ) ^{2}} = tan (f)

tan (f) = u :

على افتراض أنّ

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

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

يتحقّق لكلّ

u

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

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

اضرب الطرفين بـ

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

بسّط

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

ربّع الطرفين:

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

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

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

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

وسّع:

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

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

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

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

:فعّل قانون القوى

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

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

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

:فعّل قانون القوى

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

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

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 + u^{2})^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 1 + u^{2}

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

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

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

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

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

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

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

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

وسّع:

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

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

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

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

زوجيّ n

إذا تحقّق أنّ

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

:فعّل قانون القوى

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

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

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

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

إلغ العوامل المشتركة

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

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

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

وسّع:

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

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

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

:فعّل قانون القوى

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

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

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

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

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

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

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

وسّع:

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

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

(\frac{u}{1 + u ^{2}})^{2} = \frac{u ^{2}}{1 + u ^{2}}

(\frac{u}{1 + u ^{2}})^{2}

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

:فعّل قانون القوى

= \frac{u ^{2}}{( 1 + u ^{2} ) ^{2}}

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

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((1 + u^{2})^{\frac{1}{2}})^{2}

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

:فعّل قانون القوى

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

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

\frac{1}{2} \cdot 2

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

:اضرب كسور

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

2 :

إلغ العوامل المشتركة

= 1

= 1 + u^{2}

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

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

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

a (b - c) = ab - a c

: افتح أقواس بالاستعانة بـ

a = u^{2}, b = 1, c = \frac{u ^{2}}{1 + u ^{2}}

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

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

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

1 \cdot u^{2}

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

اضرب

= u^{2}

\frac{u ^{2}}{1 + u ^{2}} u^{2} = \frac{u ^{4}}{1 + u ^{2}}

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

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

:اضرب كسور

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

u^{2} u^{2} = u^{4}

u^{2} u^{2}

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

:فعّل قانون القوى

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

= u^{2 + 2}

2 + 2 = 4 :

اجمع الأعداد

= u^{4}

= \frac{u ^{4}}{1 + u ^{2}}

= u^{2} - \frac{u ^{4}}{u ^{2} + 1}

u^{2} = \frac{u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

:حوّل الأعداد لكسور

= - \frac{u ^{4}}{1 + u ^{2}} + \frac{u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

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

:بما أنّ المقامات متساوية، اجمع البسوط

= \frac{- u ^{4} + u ^{2} ( 1 + u ^{2} )}{1 + u ^{2}}

- u^{4} + u^{2} (1 + u^{2})

وسٌع:

u^{2}

- u^{4} + u^{2} (1 + u^{2})

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

وسٌع:

u^{2} + u^{4}

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

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = u^{2}, b = 1, c = u^{2}

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

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

1 \cdot u^{2} + u^{2} u^{2}

بسّط:

u^{2} + u^{4}

1 \cdot u^{2} + u^{2} u^{2}

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

1 \cdot u^{2}

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

اضرب

= u^{2}

u^{2} u^{2} = u^{4}

u^{2} u^{2}

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

:فعّل قانون القوى

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

= u^{2 + 2}

2 + 2 = 4 :

اجمع الأعداد

= u^{4}

= u^{2} + u^{4}

= u^{2} + u^{4}

= - u^{4} + u^{2} + u^{4}

- u^{4} + u^{2} + u^{4}

بسّط:

u^{2}

- u^{4} + u^{2} + u^{4}

جمّع التعابير المتشابهة

= - u^{4} + u^{4} + u^{2}

- u^{4} + u^{4} = 0 :

اجمع العناصر المتشابهة

= u^{2}

= u^{2}

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

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

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

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

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

حلّ:

يتحقّق لكلّ

u

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

من الطرفين

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

اطرح

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

بسّط

0 = 0

يتساوى الطرفان

يتحقّق لكلّ u

يتحقّق لكلّ u

u = tan (f)

استبدل مجددًا

يتحقّق لكلّ tan (f)

يتحقّق لكلّ tan (f)

tan (f) = u \in R

يتحقّق لكلّ

: f = arctan (u \in R يتحقّق لكلّ) + πn

tan (f) = u \in R يتحقّق لكلّ

Apply trig inverse properties

tan (f) = u \in R يتحقّق لكلّ

tan (f) = u \in R

يتحقّق لكلّ

:

حلول عامّة لـ

tan (x) = a \Rightarrow x = arctan (a) + πn

f = arctan (u \in R يتحقّق لكلّ) + πn

f = arctan (u \in R يتحقّق لكلّ) + πn

وحّد الحلول

f = arctan (u \in R يتحقّق لكلّ) + πn

arctan (u \in R يتحقّق لكلّ) + πn :

بما أنّ المعادلة غير معرّفة لـ

f \in R لا يوجد حلّ لـ

الحل للمعادلة هو

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

x = \frac{tan ( f )}{1 + tan ^{2} ( f )}

أمثلة شائعة

4sin(θ)=4(1-sin(θ))

4 sin (θ) = 4 (1 - sin (θ))

sin(2x)=((8m-2))/5

sin (2 x) = \frac{( 8 m - 2 )}{5}

csc(3x)=sin(3x)

csc (3 x) = sin (3 x)

arctan(x)=pi+1

arctan (x) = π + 1

sin^2(x)=((10m-7))/9

sin^{2} (x) = \frac{( 10 m - 7 )}{9}